The NES has a 6502 family CPU built by Ricoh (it's missing the BCD mode). 6502 CPUs have a 16 bit address space which can be addressed using various modes.
You can directly specify a 16 bit address in the instruction i.e.
LDA $8000
will load the accumulator with the data at address $8000
LDA $8000,X
will add the X register to $8000 then read the value at that address (i.e. if X is 76 then address $8076)
More interesting is the Zero page which can be viewed as 128 16 bit pointer registers. Zero page instructions only need one byte to specify the address (but need more cycles to perform the indirection)
So if the memory at $0004 contains the value $1234 and the Y register is $21 then the instruction
LDA ($04),Y
will read the $1234 from the zero page then add the Y register to get the address $1255, the contents of $1255 is then loaded into the accumulator. There are more addressing modes but that feels more like something for a tutorial on 6502 assembly.
Broadly speaking processors are classed by the width of their data path, however CPU designers might specify less data or address pins on the physical chip for cost reasons.
For example the 8088 & 8086 were classed as 16 bit devices with a 20 bit address bus and either an 8 or 16 bit data bus. The Motorola 68000 had another pin reduced version with a 8 bit bus (68008 used in the Sinclair QL) and later parts had full 32 bit address / data buses.