The best mathematicians and their important contributions???Watch
Quite a feat
Otherwise, the "classic" set would probably be along the lines of Gauss (first and foremost), Euler, pretty much the entire French Academy of Sciences in the 19th Century (including, among others, Cauchy, Lagrange, Fourier, and Poisson), Riemann (a student of Gauss) all are major figures in modern and historical mathematics. Emmy Noether deserves a special mention, as not to isolate her and make her a standard bearer for women, but she was remarkable in reaching great heights as a female academic in her time - and her work continues to underpin much a great deal of theoretical physics, and she like Galois also made signifcant and far reaching findings in algebra.
Leibniz and Newton deserve some reference as well, given that between them they created the foundations of calculus, which led to the birth of analysis and all it's related fields and is fundamental to much of modern science. Additionally the Bernoulli's are varied and many, and there are a number of notable contributions by different members of the family to mathematics.
The most notable modern mathematicians I can think of are probably Paul Erdos, one of the (if not the) most prolifically published academics in the world, whose work spanned a much larger number of areas than common for modern mathematicians (also namesake of the Erdos, and hence Bacon-Erdos, numbers). Von Neumann was quite prolific as well, and Nash is a well known example from "A Beautiful Mind" (which completely glosses over his bisexuality, incidentally). Terence Tao is still alive, and was well known as a mathematical prodigy as a child and is still a very active and well known mathematician (he also supervises PhDs at UCLA last I recall, so there's that too ). A number of those who I would otherwise include may well be labelled "physicists" and this overlapping area of mathematical and theoretical physics has a great number of fairly famous and even if not, important academics, so I won't go go into that area.
It's certainly worth noting however a great number of major results were known to Middle-Eastern and South Asian scholars even in fairly distant history - usually well before European (or more generally, "Western") mathematicians recreated these results. Sadly, the latter often are the namesakes of these results, and the only one I can think of offhand is Brahmagupta. Additionally Egyptian and Babylonian mathematicians had some results considerably earlier than would possibly be expected, although this was usually in a rougher and more applied form.
You can't go wrong with Euler though.