After the PCB is designed, what do I need to check in the Gerber files?
The main point for me about looking at the Gerbers outside my primary CAD is to make sure everything looks OK. I put a lot of trust in my main CAD package, and use the Gerber viewer as a qualitative verification.

Things I look for:


All layers are aligned
All layers are present (file exists)
All layers have data (not just vias)
Board outlines have the correct dimensions
Fill polygons have the right isolation, orphan settings
Make sure your soldermask is correct near high-density parts (tented vias, etc.)


Making sure that Eagle merges the layers correctly is my biggest worry when I'm checking the Gerbers, because if you aren't using a 100% verified CAM flow, what you see on the page might not be what you get in the Gerbers. Other than that, everything should be the same. Think of it as looking at a printer's proof before ordering a lot of copies.