This is the inside of my toilet.
See that blue disk? the one with the huge circular crack? that's what needed to be replaced. You see, when you push the flush-lever down, that black piece comes up to the blue piece, thereby unplugging the hole at the bottom of the tank allowing water to flow down into the bowl. Then, the black piece needs to go back down and plug the hole back up so this tank will fill. With the blue piece broken, that simply wouldn't happen. The black thing would get stuck up at the top and the toilet would continue to run.
So, I went to ACE and bought the piece below, an entirely new pump piece. All I needed was that piece off the top (not shown). I unscrewed it from the new one and put it where it belonged in the toilet.