I think Anand did an article on this when the AMD64 was about to come out...
When talking about x86 processors (pentium and athlon processors and some others), the bit "essentially" refers to the amount of data that can be stored in registers and therefore that the processor can address in memory. The limit of 32 bit processors is 2^32 or about 4gb. 64 bit increases this to 2^64 which is some ridiculously large number.
When talking about x86 processors, the "width" of the int registers determines the amount of memory that the processor can address and therefore the "width" of the int registers determines what "bit" a processor is. There are other registers (floating point, sse, sse2, etc...) of various sizes, however, these registers cannot "point" to different places in memory and therefore do not affect the amount of memory that a processor can address.
As for consoles, I know the processor used in the playstation 2 has 128 bit wide int registers... I think most of the others use 32 bit. However, I think, when dealing with consoles, it is more of a naming convention than something that has any real meaning. For example, most processors use 128-bit wide or larger data buses and some have 128-bit sse2 registers. These can't address 64 bits worth of memory, but I guess you can still call them "128 bit" systems...