The 6502 processor has seven internal 8 bit registers and an ALU (arithmetic logic unit).

The X and Y registers are called INDEX  registers and are used in the manipulation of data.

S is the stack register. It contains an address indicating where there is a temporary store for data called the STACK.

PROGRAM COUNTER HIGH AND PROGRAM COUNTER LOW together form a 16 bit register, which contains the address of the next instruction to be executed.

A is the ACCUMULATOR. All data to be manipulated is placed in this register and worked upon.

P is the PROCESSOR STATUS REGISTER.
It contains several one bit FLAGS.
These contain a one or a zero to indicate that something has happened during manipulation of data.
For example, one of the flags is set if the result of a calculation is zero.
Another is set if the result is negative.
By looking at these flags the cpu is able to make decisions.
This is why it is so clever.

The ALU is not a register. It is discussed on another page.