It is just a small hive with little space so that the bees can heat it and defend it easier, and thereby focus more on expanding. Any reason that they might benefit from this is a good reason to use them.
I think that "nuc" is short for "nucleus". So you can use them as the "nucleus" to start out a new full size hive, using the reasons that Two Bees described.
Plus they take up less space, use fewer frames for each nuc hive, and are easier to move.
Other advanced reasons to use might be:
Small hives to use when getting multiple queens mated and laying
Overwintering multiple hives - 4 nuc colonies versus 1 2-story full hive
If you just want to keep a couple of hives for honey, you probably won't use them. If you have extra equipment you probably don't need them. You may, however, buy a nuc when setting up a hive with bees, that is usually just 5 frames of bees/brood with a queen that forms the nucleus to the new hive.