A QT is kind of like a "mini-me" DT in some ways I suppose. Typically a QT is used as a transition tank. When you buy new livestock, you can choose to place them in QT before adding them to your DT. In QT, you can watch the new addition to make sure that it is healthy, eating well, etc. A QT provides the new livestock addition with a safe place, free of competition to adjust. While in QT, you should observe the livestock addition for dieases, pests, etc., and treat accordingly. When you feel they are ready (they're healthy and adjusted), you simply drip them into your DT.
Typically a QT is pretty bare compared to a DT. I have a 50 gallon DT, and set-up a 10 gallon QT for my new additions. My QT was just a bare-bottom tank, with a penguin power filter, and some pvc pipe to give the fish a place to hide. I've also used my old Biocube 14 as a QT (again bare-bottom with pvc).
You'll find many opinions on quarantining procedures and many opinions on whether quarantining is even necessary. Since my LFS keeps their fish in medicated systems, I feel that I don't need to QT fish from them, and so far, I have never had a problem (knock on wood). As for corals, typically I just dip them in iodine (sometimes freshwater as well) and then add them to the DT. However, in a few instances I've used QT systems (i.e. to allow my mandarin to adjust and train him from live to frozen food; for corals that had pests, etc.)
Here is an article that was given to me awhile back. It's worth reading:
An Ounce of Prevention is Worth a Pound of Cure: A Quarantine Tank for Everything by Steven Pro - Reefkeeping.com