Wiring – What’s the deal with this ground wire

It's called a bootlegged ground. This is commonly done in older houses that had a two prong receptacle and was updated to a three prong receptacle. The old house didn't have a ground and this tricks the inspector's electric checker, so your house passes inspection.

Yes, you wired it wrong. :)

The red wire is another hot leg. This is a common installation when the electrician is providing wires for a ceiling fan with lights (to be independently switched). The red wire shares a common neutral with the black wire, hence the three white wires. You should keep the white wires connected. Disconnect the red wire and cap it off. There should also be a bare copper or green wire in there (maybe screwed to the back of the box if the box is metal). This wire is your ground.

Update:

As @Isaac Kotlicky pointed out, the red wire could be the switched wire. But your switch could also be miswired. Take a look at the switch to see what colored wires are going into and out of the switch. A picture would not hurt.

As far as your safety ground is concerned, you really should have a bare copper or green wire in the box. If not, it's possible that the gfci switch is making up for the weakness of having no ground wire. Make sure you test the GFCI device once you get this together and working. Another possibility is that you have metal conduit leading to the box, and it is grounded. In either case, you could simply attach the fixture ground to the metal box.