What is the function of the push / pop instructions used on registers in x86 assembly?

pushing a value (not necessarily stored in a register) means writing it to the stack.

popping means restoring whatever is on top of the stack into a register. Those are basic instructions:

push 0xdeadbeef      ; push a value to the stack
pop eax              ; eax is now 0xdeadbeef

; swap contents of registers
push eax
mov eax, ebx
pop ebx