If you have a Grigri or similar auto locking belay device, and you need to get down a route, you have a few options.
One is to fix one strand of the rope, typically by putting a clove hitch at the master point, attach your Grigri, and rappel on that single strand.
Another option is this: with the rope through the anchor, tie in to one end of the rope, attach to the other side of the rope with your Grigri, and a lower yourself. This option might be better if you have a skinnier rope and/or a newer sheath, because the friction of the rope passing through the master point will slow you down a bit and possibly give you better control.
A couple of safety precautions:
Be absolutely 137% sure the rope is going through some kind of METAL connection (carabiner, quicklink, chain, rap ring, etc) at the anchor point, never webbing!
Tie a knot in the end of the rope, or maybe better yet clip the other end of the rope to your belay loop with a locking carabiner to close the rope system, so there is no chance you can rap off of the end.
See this diagram below from Petzl, it pretty much sums it up.