Memory addressing capabilities are determined by the processor. A 32-bit processor can only use up to 32-bits to represent an address. That means it can address 2^32 (4,294,967,296; 4GB) distinct bytes. But the memory is managed by the OS. So it's possible to have an OS that won't use more than 4GB even though the processor can handle more than that.

Also, most modern x86 processors have a hack - Physical Address Extension (PAE) - that let the 32-bit processors address up to 64 GB.