Refers to processors where integers and other data are stored in 32-bit registers. This has been the dominant processor architecture for a while, only recently being superseded by 64-bit operating systems.
The maximum integer value that can be stored in 32 bits is 4,294,967,295. Therefore, a processor with 32-bit memory addresses can directly access "only" 4 GB of memory. Which is why it doesn't make sense to put any more memory in a system running a 32-bit operating system.