What are Registers?
The processor can operate upon numeric values (numbers), but these have to be stored somewhere first. The data are stored mostly in memory, or inside the instruction opcode (which is stored usually in memory too), or in special on-chip memory placed directly in processor, which is called register.
To work with value in register, you don't need to address it by address, but special mnemonic "names" are used, like for example
ax on x86, or
A on Z80, or
r0 on ARM.
Some processors are constructed in a way, where almost all registers are equal and can be used for all purposes (often RISC group of processors), others have distinct specialization, when only some registers may be used for arithmetic ("accumulator" on early CPUs) and other registers for memory addressing only, etc.
This construction using memory directly on the processor chip has huge performance implication, adding two numbers from registers storing it back to register is usually done in shortest possible time by that processor (Example on ARM processor:
ADD r2,r0,r1 sets
(r0 + r1) value, in single processor cycle).
On the contrary, when one of the operands is referencing a memory location, the processor may stall for some time, waiting for the value to arrive from the memory chip (on x86 this can range from zero wait for values in L0 cache to hundreds of CPU cycles when the value is not in any cache and has to be read directly from memory DRAM chip).
So when programmer is creating some data processing code, she usually wants to have all data during processing in registers to get best performance. If that's not possible, and memory reads/writes are required, then those should be minimised, and form a pattern which cooperates with caches/memory architecture of the particular platform.
The native size of register in bits is often used to group processors, like Z80 being "8 bit processor", and 80386 being "32 bit processor" - although that grouping is rarely a clear cut. For example Z80 operates also with pairs of registers, forming native 16 bit value, and 32 bit 80686 CPU has MMX instructions to work with 64 bit registers natively.