At this link, you can find a loose translation of „Die Entwicklung der Mathematik an der Universität Göttingen" by Erwin Neuenschwander and Hans-Wilhelm Burmann, an interesting article published in the book Die Geschichte der Verfassung und der Fachbereiche der Georg-August-Universität zu Göttingen.
In the early times, Segner, Penther and Tobias Mayer played an important role in the development of Göttingen as a relevant center of mathematics and science in Germany:
At the founding of the university, only five chairs were planned in the Philosophical Faculty. Its first mathematician, appointed in 1735, was Johann Andreas von Segner, who had studied philosophy, mathematics and medicine in Jena and had practiced as a physician before starting his academic career. Apart from mathematics, Segner was responsible for the lectures in physics, wrote numerous textbooks on both subjects, and occasionally lectured on chemistry. His prolific activity soon brought Segner a reputation as one of the best living mathematicians in Germany. Friedrich Penther, originally hired as overseer of the academic buildings in Göttingen, was appointed professor for mathematical topics in areas such as economics, architecture, and mapmaking. After Penther’s death his place was occupied by the well-known astronomer and mapmaker Tobias Mayer. Already before Mayer’s arrival in Göttingen, Segner had campaigned for the establishment of an astronomical observatory, which he and Mayer then shared for some time; but as competition over the observatory grew and the authorities in Hanover generally supported Mayer, Segner left Göttingen in 1755 to take over the chair vacated by Christian Wolff at the University of Halle.
Once Segner left Göttingen, Kästner, his successor for almost 50 years, was defined by the German physicist Georg Christoph Lichtenberg in one of his aphorisms as "an encyclopaedic dictionary":
Segner's successor was Abraham Gotthelf Kästner, who worked in Göttingen for almost 50 years. When not busy with mathematics, he worked especially in physics and astronomy, as well as in chemistry, botany, anatomy, philosophy, literature, and many other areas, and served as director of the observatory after Mayer’s death. In one of his now classic aphorisms, Georg Christoph Lichtenberg called Kästner "an encyclopaedic dictionary." One of his many influential books was the four-volume Anfangsgruende der Mathematik, or "starting points of mathematics," arranged by field and treating mechanics and hydrodynamics in addition to arithmetic, geometry, analysis, algebra, and applied mathematics. In Kästner's time Albrecht Ludwig Friedrich Meister and Mayer's brother-in-law Georg Moritz Lowitz taught applied mathematics, Lichtenberg and Johann Christian Polykarp Erxleben taught physics, and Karl Felix Seyffer taught astronomy.
But, after Kästner's death, there was a period of crisis until the arrival of Gauss:
[...] no suitable replacement could be found, and for a time there was no towering personality of the kind the University of Göttingen would find some years later in Gauss. At the same time, Hanover experienced a political crisis, submitting to Prussian and then French occupation.
Then, of course, as mentioned in other answers, there was Gauss, the most important mathematian of his time and one of the most relevant astronomers and physicists in Europe:
Carl Friedrich Gauss came from Braunschweig, or Brunswick, where he spent his youth. Thanks to a scholarship from the duke of Braunschweig, who recognized and supported his talent, he went to Göttingen in 1795 to study mathematics there. He came into particularly close contact with Seyffer, the professor of astronomy at the time, and thought highly of Lichtenberg, but was less impressed by Kästner and the other mathematics lecturers. Other German mathematicians, however, may not have been able to meet his standards either. In 1798 Gauss returned to Braunschweig, where he worked on his magnum opus Disquisitiones Arithmeticae and received a doctorate in 1799 in Helmstedt. With the publication of the Disquisitiones and the sighting of the dwarf planet Ceres based on his computations in 1801, Gauss became one of the most prominent astronomers and mathematicians in Europe. He received a salary increase from the Duke of Braunschweig and an offer to direct the observatory in St. Petersburg, upon which Göttingen Professor Wilhelm Olbers stepped into negotiations with the University of Göttingen to do as much as possible to keep Gauss in Germany. Among Gauss’s main reasons for accepting the resulting appointment in 1807 were the Hanover government’s plan to establish a new observatory in Göttingen and the relatively light obligations which he had to take over within the university. After the completion of the new observatory in 1816, Gauss moved in as its director and lived there until his death in 1855.
Gauss was the leading mathematician of his time, and many regard him simply as the greatest mathematician. At 18 he discovered that the regular seventeen-sided polygon can be constructed with a ruler and compass, the first major discovery in this area in 2000 years. His Disquisitiones Arithmeticae provided the foundations for modern number theory. His methods for the calculation of planetary motion have still not been essentially improved upon even today. Just how far Gauss was ahead of his contemporaries in complex analysis, and especially in his insights about non-Euclidean geometry, became clear only with the publication of his posthumous works, letters, and scientific diary. His interest in geodesy —he started a project to survey the lands of the Kingdom of Hanover in 1818 and carried its main workload for some years— led him to the investigation of curved surfaces, culminating in the Theorema Egregium and other results developed further by Riemann. Together with William Weber he investigated electricity and magnetism, leading to the invention of the electrical telegraph as a byproduct of their research. These are only a few of the highlights of Gauss's enormous output. He made crucial discoveries in almost all areas of mathematics, and shaped the entire field with his demands for mathematical rigor.
At Gauss's time, also Thibaut played an important role as a prominent lecturer:
Gauss seems to have had an aversion to lecturing at least in his earlier years, and appeared content when his lectures were cancelled due to a lack of students. In Gauss's time, basic mathematics lectures were conducted largely by Thibaut, mentioned above, and later by George Karl Justus Ulrich and Moritz Abraham Stern. Thibaut usually covered pure mathematics, differential equations and integrals, and finite analysis, while Ulrich lectured on solid geometry and trigonometry, applied geometry, mechanics and civil architecture. Thibaut was known as the best lecturer in Göttingen, and because of his perfected rhetorical style was even compared with Goethe.
After Gauss's death, there was again a period of crisis, which seemed to have arrived to its end with the arrival of Dirichlet. But, his premature death in 1859 prevented him for working in Göttingen for a long time. Riemann succeed Dirichlet, but he also died from tuberculosis in 1866, so, once again, Göttingen remained without a leading figure:
The death of Gauss left the University of Göttingen with a gap that could not easily be filled. The directorship of the observatory remained vacant for several years and was administered intermittently by Weber until 1868, when it was finally divided between William Klinkerfues and Ernst Schering. In the mathematical areas a worthy successor was found in Peter Gustav Lejeune Dirichlet, who at that time enjoyed the highest reputation among German mathematicians. Dirichlet's work united two currents, number theory in the tradition of Gauss and applied mathematics from the French school, as for example in his theorem on prime numbers in arithmetic progressions together with its beautiful proof, and in the extraordinarily fruitful Dirichlet principle. Dirichlet was an inspiring teacher, and his lectures on number theory, published after his early death by Richard Dedekind, remained the standard work for decades.
Dirichlet's successor was Bernhard Riemann, who had studied first in Göttingen and then under Dirichlet in Berlin and obtained his doctorate and Habilitation under Gauss in 1851 and 1853. It was a great moment in mathematics when the young Riemann delivered his Habilitation lecture "On the Hypotheses that Lie at the Foundations of Geometry" in front of Gauss; Gauss was deeply impressed by it and, according to Weber, had the highest expectations about him, and Riemannian geometry would later supply the mathematical framework for Einstein's relativity theory. Riemann's innovative ideas were understood in their full depth only gradually, but their effects were lasting and provide mathematics with essential stimulation even today. He set the theory of analytic functions on a firm foundation and gave it a new dimension with the Riemann mapping theorem, which he proved with the help of the Dirichlet principle. The notion of Riemann surface shed light on the preceding half century's investigations of algebraic curves, unified and standardized them and opened the way for many future developments in algebraic geometry and topology. In an eight-page work on number theory, Riemann provided the key to problems of prime number distribution. Hilbert remarked in a lecture (W.S. 1896/97, p. 264) that "only very rarely has an essay of such shortness, sharpness and genius flowed from the pen of a human being, as did this masterwork of one of the greatest spirits of our science." The Riemann hypothesis, whose demonstration would greatly extend our knowledge of the distribution of prime numbers, has defied all attempts at proof and refutation until today.
Unfortunately, Riemann, like Dirichlet, was prevented from working for a very long time as a full professor in Göttingen, since he contracted tuberculosis just three years after his appointment and then spent much of his remaining time at health resorts in Italy. After Riemann came Alfred Clebsch, who also died a few years after his appointment, and then Lazarus Fuchs, who one year later followed a call to Heidelberg. For a brief period Göttingen mathematics once again lacked a long-term leading figure, and ceded supremacy to Berlin, just as the nearby capital Hanover ceded much of its political power to Berlin around this time.
Then, there was the time of Schwarz:
After the departure of Fuchs in 1875, Hermann Amandus Schwarz was appointed from Zurich. Schwarz organized his lectures according to a well worked out study plan and arranged them into two cycles. The first, introductory cycle covered differential and integral calculus, analytic geometry, surfaces of second degree, curved surfaces and double curvature, and synthetic geometry. The second, simultaneously running cycle covered analytic functions, elliptical functions, minimal surfaces, the hypergeometric series, and other areas of function theory. Schwarz, supported by Stern, spearheaded the creation of a circulating library for the Mathematics-Physics Seminar in 1878, and it remained under his administration until his departure for Berlin.
And, as explained in Mikhail Katz's answer, Klein played a relevant role as a dynamizer and a great organizer of mathematics and physics in Göttingen:
Felix Klein, the great organizer of Göttingen mathematics and physics, studied in Bonn, Göttingen, Berlin and Paris, particularly with Plücker and Clebsch, and received his Habilitation in Göttingen in 1871. In 1872 he was appointed full professor at Erlangen, from where he went to Munich in 1875, to Leipzig in 1880, and finally back to Göttingen upon Stern's retirement in 1886. In his 1872 Erlangen Program Klein used the notion of group to formulate a classification principle for geometry that had a lasting effect on geometrical thought. He saw himself as a developer of the brilliant ideas of Riemann, whose geometrical core he worked out further and brought into the investigation of model functions and automorphic functions.
Klein's main objectives apart from pure mathematics were the reinforcement of the connections between mathematics, the natural sciences and technology and the restructuring of education in mathematics and science from the earliest grades to the university. Both of these goals shaped much of his time in Göttingen. Already in his appointment negotiations, Klein advocated the creation of a mathematical reading room with a reference library like the one he had organized in Leipzig. His request was approved by the time he came to Göttingen, and immediately after his arrival the Reading Room of the Mathematics–Physics Seminar was opened. At that time it was in Auditorium No. 20 on the upper floor of the auditorium building at Weender Gate, directly beside the model collection and the lecture rooms of the Mathematics–Physics Seminar. At first its furnishing was relatively modest, with 20 working places and a library of about 500 volumes around 1890.
With Schwarz's departure for Berlin as successor to Weierstrass in 1892 and the subsequent departure of Heinrich Weber, Klein had the freedom to reorganize mathematical instruction in Göttingen according to his wishes. The small seminar library developed by Schwarz was combined with the library of the mathematics reading room and put under Klein's direction. Klein introduced term fees for the gradually expanding student body and applied for 3000 Marks for the library's further development, a sum five times the annual budget for mathematics. He also took over the collection of mathematical instruments and models and obtained funding for a special assistant for it, and took part in the establishment of the Mathematical Society and the first edition of study plans, distributed to students free of charge upon matriculation.
At the same time, Klein worked on his second main goal: strengthening the ties between mathematics, the natural sciences and technology. Gathering together interested professors and industry leaders, he created the Göttingen Association for the Advancement of Applied Physics and Mathematics in 1898. This association raised over 200,000 Marks over the next ten years to support these sciences in Göttingen, allowing for the establishment of numerous new buildings and institutes. At the 10-year anniversary celebration, Klein could report that since the Association’s inception the Institutes for Applied Electricity, Applied Mathematics and Mechanics and the Geophysics Institute had been founded, and the number of professors in mathematics and physics had doubled. This led, among other things, to the appointment of Hermann Theordor Simon, Carl Runge, Ludwig Prandtl and Emil Wiechert.
The number of mathematics students and reading room users was constantly rising, and space shortages and the move of the Physics Institute to Bunsenstrasse left the mathematics and physics facilities scattered across the city. Klein felt a growing need for a new Mathematics Institute. By 1911 he believed this dream almost attained, as the Göttingen Association had allocated 200,000 Marks for the purpose and its chairman had purchased a suitable property on Bunsenstrasse in direct proximity to the physics institutes. But the realization of Klein's plans was delayed by the First World War and later by the struggling economy and spiralling inflation, and one of his great projects did not come to fruition in his time.
Klein attracted Hilbert to Göttingen and both of them brought Minkowski and Landau:
In the meantime, however, Klein continued to further mathematics in many other ways. He served for several years as chairman of the International Commission on Mathematical Instruction and oversaw several series of publications associated with the Commission. He served as editor of the leading mathematics journal Mathematische Annalen and the monumental Encyclopedia of the Mathematical Sciences and their Applications. Among his greatest victories for Göttingen was his success, thanks to his excellent relations with Prussian Ministry Director Friedrich Althoff, in drawing David Hilbert to Göttingen in 1895 and in keeping him despite repeated calls to other universities. After one such competing offer in 1902, Hilbert and Klein convinced Althoff to create a third mathematics professorship in Göttingen, which was then occupied first by Hermann Minkowski and then by Edmund Landau. Thus Göttingen mathematics rose to first place among German universities even in the number of professors.
The mathematical center of Göttingen at that time was undoubtedly David Hilbert, who shaped 20th century mathematics as no one else has. Hermann Weyl wrote in a letter in 1927: "The spirit in which we do mathematics, we received from him." In his universality —every few years he turned to a completely new sphere of activity— he is comparable only with Gauss. He ended the classical era of invariant theory with his new "transcendental" methods. His unification of algebraic number theory, commonly known as the Zahlbericht or "report on numbers," was for at least a half century the classic work for everyone concerned with the field. His Foundations of Geometry inaugurated the axiomatic method. He introduced a method for the justification of the Dirichlet principle which now belongs to the essential tools of analysis. In his investigations of integrals he recognized the necessity of treating infinite-dimensional areas —one speaks today of Hilbert spaces— and he developed a spectral theory that provided the foundations for quantum theory and gave rise to functional analysis, a powerful branch of modern mathematics. His program for the proof of the consistency of mathematics could not attain its goal, as Kurt Gödel showed in 1933, but it directed attention to mathematical models of calculating machines and to the theory of formal languages, which today form the basis of computer science and computer engineering. In his famous lecture at the Second International Congress of Mathematicians in Paris in 1900, Hilbert put forth a now classic list of 23 unsolved problems in a range of areas of mathematics, setting the agenda for much of 20th century research. The overall course of his lectures is of tremendous span, covering all areas of pure mathematics and extending to physical topics such as mechanics, electromagnetic oscillations, and relativity theory, and to philosophical topics such as the nature of mathematical knowledge. He attracted many students, and among his 69 graduate students were eminent figures such as Otto Blumenthal, Richard Courant, and Hermann Weyl.
And, from these period, Göttingen became an attraction pole for many of the most relevant mathematicians and physicists of the time.