How can I take the pop-up stopper out of my sink?
Grip the nut with the 2 'slots' and turn it counter-clockwise until it spins off and you can pull the pivot rod backwards enough to release the plunger. Reverse the process to put it back together.

A plastic setup like that is designed to twist off by hand. If it seems too tight simply use a pair of pliers to grab it and turn it.

Good luck!