In the finale episode, "Fast Enough", or Season 1 Episode 23 Eddie shoots himself in the heart in order to stop the Reverse Flash. But, later in season 2, we see the Reverse Flash gets trapped in the past again. How can he be back if Eddie is gone?
This Reverse Flash is a time remnant. I don't believe it's explicitly stated how this time remnant was created, but it's possible that Barry caused it when he pulled him out of the timeline and created Flashpoint. Harry says:
It's possible Eobard was in the Speed Force, protecting him like a bomb shelter, keeping him alive and his timeline intact. It's what's known as a timeline remnant.
...there is nothing I can do about it. There's nothing you can do about it. This is his origin story, and it's going to happen no matter what you do.