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What is the significance of a processors bit capacity?

I'm learning about how processors send an address to ram through the address bus, and receive and send data from ram through the data bus. In the videos I have watched, the processor was an 8 bit processor that had 8 wires coming out for the data bus, and 8 wires coming out for the address bus, where each wire carried one bit of information, either off or on. That got me wondering. Does a 32 bit processor basically work the same way, but it can send more bits through the address/data bus at one time. What about a 64 bit processor. I haven't learned about registers yet, but I'm getting there. Does it have something to do with the size of the registry? I understand the parts of a processor, but I don't understand what people mean when they classify a processor by its bit capacity.