A stack is a way for a CPU to keep track of the order of execution when it calls into functions. There is an instruction (CALL) that will cause the CPU to begin executing instructions in a different region of the program. When the program reaches another instruction (RET), it must be able to go back to where it was when it hit the CALL instruction. The stack holds the PC value when the CALL instruction was executed, and the RETURN statement pulls that address from the stack and puts it back into the PC so the CPU will execute it on the next cycle.
The CHIP-8 has 16 levels of stack, meaning it can hold 16 different PCs. Multiple levels allow for one function to call another function and so on, until they all return to the original caller site.
Putting a PC onto the stack is called pushing and pulling a PC off of the stack is called popping.