What is a Nuc (pronounced 'nuke', short for Nucleus Hive)? Generally, a beekeeper is referring to a small hive body that holds only 5 frames. It is about half the size of a Medium or Deep Hive body. This is where you start a new hive with a new queen. You give them a small cavity of space to maintain, which makes it easier for them to get a good start and monitor the entire area of their home.
There are a couple of ways to create a Nuc and several different ways to house them.
With a New Queen
The Virginia State Apiarist (Keith Tignor) recommends this as the best way. You purchase a (or raise your own) queen, create the Nucleus and introduce the new queen to it. You start right off with a laying queen.
Without a New Queen
This is the method that I like for most beekeepers. I honestly believe that you want to be selecting bees from your own apiaries (unless all of your hives are doing poorly.)
The main three things that you want for any Nuc are Nurse Bees, Capped Brood and Resources. The Nurse Bees will be a big help to building a new queen or helping an existing queen get her eggs converted to larva. I shake at least a frame of bees into a Nuc to ensure a healthy population for my Nuc, sometimes two shakes. Capped brood is important to provide a stready stream of new bees while you get your queen raised. Resources are mainly important early on (when a cold snap might stop by bees from foraging) and when you want your bees to raise a new queen (hungry bees do not make the best queens.)
Nucs without Swarm Cells
One way (and I personally like this method a lot) to get a new queen for your Nuc is to simply bring over a frame of eggs. In this case, I usually bring over a frame and a half or two of capped brood and then one with eggs (which are usually accompanied by larva and capped brood.) Once you get the Nurse bees into the Nuc (and they can be from several different hives - they will not fight), they will immediately realize they no longer have a new queen. They will begin raising new queens in earnest.
For this reason, make sure to include vertical eggs (eggs standing straight up in the bottom of the cell.) These eggs are 12 hours old or less. If possible, leave the Nuc in the original hive's location so that the field bees come back to arm the newly created, queenless Nuc (if this happens, I definitely only shake one frame of nurse bees into the Nuc, because they will not be lacking for a population).
For all Nucs that are moved to another location, it is wise to use a Robber Screen by mid-May or so. These Nucs will start out with few defenders/foragers and need some peaceful time to get up and going. Put the robber screen on at night, when you know that no other foreign bees are in the Nuc. If you put the Robber Screen on in the day, there is a chance that a foreign bee will be in the Nuc. She will have to navigate the Screen to get out of the hive and can communicate this back to her comrades.