Who created topology, when did that happen, and how was it discovered?
Many sources, including this one, credit the idea of topology (and its applications) to Leonhard Euler, to solve the puzzle of the Seven Bridges of Königsberg (or, rather to prove that there was no solution). Euler first used the concept in the first half of the 18th century:
Perhaps the first work which deserves to be considered as the beginnings of topology is due to Euler. In 1736 Euler published a paper on the solution of the Königsberg bridge problem entitled Solutio problematis ad geometriam situs pertinentis which translates into English as The solution of a problem relating to the geometry of position. The title itself indicates that Euler was aware that he was dealing with a different type of geometry where distance was not relevant.
Euler went on to make subsequent developments in the fledgling field of topology. The page gives essentially a complete history of topology, from Riemann to Poincaré, which might answer any other questions you have. Britannica confirms those facts regarding Euler.
This page also credits Euler with creating the concept of topology, using it to solve the Seven Bridges problem.
One of these areas is the topology of networks, first developed by Leonhard Euler in 1735. His work in this field was inspired by the following problem: The Seven Bridges of Konigsberg
Here we find an interesting PDF on the development of algebraic topology. McCleary credits Poincaré with some of the earliest work in the field, including applying topology to manifolds, although Hilbert later extended Poincaré's ideas.
Some topological problems were considered long ago, for example by Euler (see the previous answer). Some ideas about topology were even earlier proposed by Leibniz. The famous kindergarten problem about three houses and three wells belongs to this class. One notable invention was Möbius band, another Gauss knot invariant.
However, one can agree with Dieudonné's opinion that topology as a separate area of mathematrics begins with Riemann. Originally it was known as Analysis Situs, the term which existed until the beginning of 20th century. The term "Analysis Situs" was introduced by Leibniz(!)
The word topology was used by the Germans and since the beginning of the 20th century it is accepted. Depending on the point of view one may also consider this whole period before Poincaré as "pre-history", however some basic results about closed surfaces were proved at that time, and Betti introduced "Betti numbers".
The modern subject was really defined by the work of Poincaré in the very end of 19th century with his work called Analysis Situs in six parts.
References: Dieudonné, Abrege d'jistoire des mathematiques, 1700-1900, vol. 2. Poincare, Analysis situs, J. Ecole Polyutechnique 1995, 1, 1-121.
All this concerns what is called today "Algebraic topology". A different part of mathematics, called "General topology", has a different origin: in real analysis (Cantor).
Let me add another piece of history going slightly back in time.
The idea that polyhedra satisfies the formula $V-L+F=2$, where $V$ is the number of vertices, $L$ is the number of sides and $F$ is the number of faces, goes back to Descartes.
The formula is nowadays called Euler's formula (yet again, Euler!) because he was the first one to attempt a proof. I say "to attempt" because his proof was "almost" correct. But exactly in this almost layed the interesting part.
In the nineteenth century various mathematicians proposed example of "polyhedra" that were not satisfying the formula: a cube with inside a cubic cavity, a smaller cube sitting on the side of a larger cube (which satisfies or not the formula dependind on how you count faces) and so on.
I wrote "polyhedra" because the usual objection to such examples was to refine the notion of polyhedra as to exclude them.
Incidentally it is exactly in studying this kind of anomalies that Listing and Möbius came across the Möbius band.
When it turned out the the definition of polyhedra became so complicated to counter intuition, it solwly became clear that such anomalies could in fact bring in new informations... it is also in this way that the Euler-Poincaré (genus) invariant was born. A discover which sits at the core of further developments of homotopy thoery and later on homology theory.
As Alexander wrote general topology was born more in connection with clarifying the foundations of real analysis. I just want to stress that also in this case a notable role was played by "monsters", i.e. by all sorts of strange, intuition defying math objects that were developed throughout he second half of the nineteenth century and the first half of the twentieth (Cantor set, Sierpinski gasket ecc...)