A lot of the claims made in this thread about CPU architecture are faulty at best, and completely wrong at worst.
But this isn't a CPU class, so I won't get into memory addressing and other things where the "bit" value actually has significant meaning.
A CPU could be 256-bit, but still only run at 1mhz.. the result? a very slow 256-bit CPU.
Likewise, a CPU could be 8-bit but run at 3ghz. the result? a very fast 8-bit CPU.
The "bit rating" of the CPU does not determine how fast the CPU will process data. It does, however, determine how much data it can "see", and how much data it can manipulate per cycle.
TYPICALLY as people developed more modern CPUs, they also made them more capable of handling larger amounts of data as well as being able to handle them faster.
for our purposes here, we usually talk about bits in terms of color depth; 4-bits is a selection of 16 possible colors. 8-bits is 256 colors, 16-bit is 65k colors.. 24-bit and 32-bit are millions of colors, and if you go much above 32-bit you would have more color depth than the human eye can differentiate.
However, even though the NES is an 8-bit system, the CPU architecture determines how many colors the CPU can use when selecting an on-screen palette. The NES CPU architecture allows for only 4 colors per sprite, and only 16 simultaneous colors per screen.. But it can select 16 colors from a palette of 256, however.
For the SNES, it could have 256 colors on screen simultaneously, and each sprite could have 16 colors, selectable from 64k different combinations.
That's usually the most visible aspect of the "bits" a CPU has. "Under the hood", so to speak, it affects how much data the CPU can see (whether it could see 256kb of ram, or 1024kb of ram, or 8mb or 64mb or whatever).... that limitation determines the complexity of the graphics (you can't have photorealistic models stored in 256kb of memory!), the quality of the sound (which would be based on the SPU, not the CPU), the size of the game, etc.
Carts were nice, because if you needed additional performance in a game, you could just add a chip to handle specific functions, like voices, 3d positioning/rendering, or other tasks. You can't just 'add a chip' for a cd-rom, though..
I guess it would be possible to make a FPGA CPU and 'configure' it every time you put in a game.. but it would never be as efficient as a real CPU (RISC or otherwise).
Anyway... it's not what the original poster was actually asking about... but if he/she wants to know what bits REALLY do, with regards to the CPU, he/she needs to go to a university where they teach Computer Engineering, and take a course or two in CPU design.
PS: I'm not a computer engineering student. Someone who knows assembly (blip_) could do a much better job of explaining than I.