Do I have a non-grounded light fixture?
Yes, the fixture is currently not grounded, but it should be. The manufacturer has included a grounding wire because the fixture needs one to be safe and UL compliant. 

Assuming there is a ground connection in the ceiling box (either a ground wire or a box grounded through an armored cable), this can be easily fixed. The power is turned off, the bare wire threaded through the chain, the canopy is lowered, and the wire is connected to a ground wire or grounded to the fixture box itself, and the canopy is replaced.

If this is an older home with no ground wiring, it may not be possible to ground the fixture. In that case, you are no safer or more at risk than our parents were before grounding became the standard. In that case, you may still wish to thread the wire up to the box. In that way, if you ever rewire the house or move the fixture to a location with a ground in the box, it is still intact. Just make sure the ground wire does not accidentally touch a live terminal in the box.

Your question sounds as if you have not had much expereince with wiring. If that is the case, you may want to enlist a more knowledgeable friend to help, or call a professional.