How to make PNG-quality animations in GIF format?

I've noticed more and more sites using animated GIFs that have PNG quality. You'd think they're the latter if you didn't check the image info. The lines are crisp, the colors look great, there's none of that slight-but-still-noticeable graininess that GIFs get. How are people accomplishing that? I'm not a graphics designer but I'm curious and I'd like to create such animated GIFs that have no recognizable loss in quality compared to GIFs. (I use GIMP but I do have access to Photoshop, if required.)