Let me begin by saying that I agree that the question should be addressed carefully, I intend this as a pedagogical exercise. Every musician has its favorite musicians of all time, even if he agrees that one cannot make a ranking among them and it depends a lot of each person's taste. In the same fashion, I believe that we mathematicians are allowed to admire great mathematicians. This can be useful to inspire current and future generations. I don't want a ranking nor anyone to say who is the best. But to their opinion and KNOWLEDGE of mathematics, what recent achievements led to new ways to conceive mathematics.
My B.A. thesis supervisor, who is a formidable mathematician, told me once that there weren't any contemporary mathematicians as great as those of the past (by past, meaning dead before the 1960's or something like that). He is a great fan of Henri Lebesgue for who he already considers a mathematician of the old golden years. The criteria one can use to measure greatness is the size and variety of their mathematical legacy.
There are many examples of amazing achievements such as the proof of Fermat's last theorem or the Poincare's conjecture via the Geometrization Conjecture. Nonetheless, mathematical legacy is not the same as great technical achievement. Clearly mathematics needs both of them but in this case, however, I am more interested in the creation of new and unexpected areas of mathematics.
I want to make a list of counterexamples since I believe the statement is mostly false. My personal favorite counterexample is William Thurston whose powerful insights helped to bring together the fields of geometry and topology. What are other great examples and why?
I believe this question is important since most undergraduate students actually ignore this facts. By having a clear understanding of who are the greatest new mathematicians and the subfields that emerged after their work then undergraduate students will have a better sense of direction and motivation to begin their graduate studies. This is at least true in my country were few seminars and talks regarding the current state of mathematics are given. This is a delicate issue of course since most new problems or research topics are hard to expose in simple undergrad terms, yet, it can still be very motivating.