What do 64-bit and 32-bit mean for your computer?

I'm curious, could you please explain what 64-bit and 32-bit mean for a computer system? Specifically, how do they differ in terms of performance, capabilities, and compatibility with software and hardware? As a user, how can I tell which one my computer runs on, and is it possible to upgrade from 32-bit to 64-bit? Also, what are the advantages and disadvantages of each, and how do they impact the overall user experience?