He made fundamental contributions to mathematical logic, mathematical analysis, and partial differential equations. In particular in solving Hilbert's first problem, "Cantor's Problem on the Cardinal of the Continuum," in which he proved that the axiom of choice and the generalized continuum hypothesis are independent of Zermelo Fraenkel Set Theory, he introduced the powerful technique of forcing, which today remains one of the basic tools in set theory.
Here's a quote from Cohen's Set Theory and the Continuum Hypothesis (p. 151):
A point of view which [I feel] may eventually come to be accepted is that [the continuum hypothesis] is obviously false. The main reason one accepts the axiom of infinity is probably that we feel it absurd to think that the process of adding only one set at a time can exhaust the entire universe. Similarly with the higher axioms of infinity. Now [creating] the set of countable ordinals ... is merely a special and the simplest way of generating a higher cardinal. The set C [the continuum] is, in contrast, generated by a totally new and more powerful principle, namely the power set axiom. It is unreasonable to expect that any description of a larger cardinal which attempts to build up that cardinal from ideas deriving from the replacement axiom can ever reach C. ... This point of view regards C as an incredibly rich set given to us by one bold new axiom, which can never be approached by any piecemeal process of construction. Perhaps later generations will see the problem more clearly and express themselves more eloquently.