I'm trying to have lights on a spinning object from a normal 120v outlet. What kind of connector do I use to keep the wiring from getting tangled up?

Use a device that they call slip rings. They consist of basically two componets. 1) the slip ring that is connected to the moving or turning object. 2)The other component is mounted stationary adjacent to the slip ring and is referred to as the brush holder. These come in different configurations, but the principal is the same. The stationary brush holder has brushes that rest on the rotating copper rings of the slip ring. Power is applied to the brush holder- terminals and power is taken from the slip ring terminals mounted on its bottom or top.

Imagine a 3" diameter disk made of a solid insulation material and about 2" thick.
Embedded in the circumference of the disk are two flat copper rings spaced 1" apart.
the rings were installed in the disk with connection points that come out on the top or the bottom flat side of the disk. This is called the slip ring. Now for this we take and machine a 1" diameter hole in the slip ring which will slide onto our 1" rotating shaft.
The device on the shaft that you want to power up is connected to the terminals of this slip ring. If you turn the shaft the slip ring turns with the shaft, so no wires get twisted. To get power to these two copper rings on the slip ring you could just hold a hot wire and a neutral against the rotating copper on the slip ring. The hot on one and the neutral on the other. Don't do this! Of course you wouldn't want to do this for a number of reasons including safety and convenience, so you install two separate brush holders with carbon brushes in them and design it so the brushes have constant pressure on the slip rings. Then you connect up the power to the terminals of the Brush holders. The power goes through the brush holders through the brushes to the copper slip rings and out of the terminals on the slip ring to whatever it is you need to power. These devices are generally quite expensive and if voltage and current are above the typical low voltage signals, they need to be protected from contact and electrocuting someone.
I used these and worked with these for a lot years on a number of applications. One good example is an electric mining shovel. They run a high voltage cable to the base of the shovel, but to get it up into the rotating part of the shovel they have to have a way to keep the shovel from twisting the cable in half. This is the principal of how they do it. In the center of the shovel is what we referred to as a "Chiristmas Tree" but it was nothing more than a bunch of brush holders connected to the incoming power and a big slip ring with multiple rings that mated up with the brushes. The brushes wear out and form time to time need to be replaced. If you look at the world around you, you will notice many things that use them. A Ferris wheel, a carousel, some motors, controls inside of the steering column on some automobiles. The list goes on. I don't know your application, but you probably need to be prepared to spend some money. Good Luck and don't hold the wires like I used in the example.

[QUOTE=Studs ad;3640365]Use a device that they call slip rings. They consist of basically two componets. 1) the slip ring that is connected to the moving or turning object. 2)The other component is mounted stationary adjacent to the slip ring and is referred to as the brush holder. These come in different configurations, but the principal is the same. The stationary brush holder has brushes that rest on the rotating copper rings of the slip ring. Power is applied to the brush holder- terminals and power is taken from the slip ring terminals mounted on its bottom or top.

Replace Sliprings with Mercotac Model 230 (http://www.mod-tronic.com/Mercotac_230.html)

Selected this at random, but there are a bunch of manufacturers and styles of these type of devices.

Look at a disco ball. You may be able to adapt something like that. It is, as Studs ad noted, a slip ring contactor