Why did Zariski feel the need to define his famous topology? Was this notion used in one form or another prior to him in algebraic geometry?
Zariski introduced his topology in this paper: The compactness of the Riemann manifold of an abstract field of algebraic functions, Bull. Amer. Math. Soc., 50 (1944), 683-691. You can read it online here. See also p. 79 of his talk at the 1950 ICM: The fundamental ideas of abstract algebraic geometry, pp. 77-89 in "Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950", vol. 2, Amer. Math. Soc., Providence, R. I., 1952. Scroll down to the volume 2 link for the 1950 ICM here.
Some pages about the importance of the Zariski topology in algebraic geometry are this MO page and MSE pages here, here, here, and here. The idea that maximal ideals (not yet prime ideals) in a commutative ring could be profitably viewed as the points in a space can be seen in Gelfand's work in the 1930s, and whether or not that might have had an influence in algebraic geometry was asked on the MO page here.