A tonearm is part of a turntable. The job of the tonearm is to carry sound from the cartridge to the turntable itself, where it would be output into the connected cables.
Tonearms are usually made out of a very light material, such as aluminum, so that accuracy is increased when playing back vinyl. Lighter tonearms also put less weight on the stylus, thereby reducing wear of the record. The downside to this is that it increases the amount of skipping.
Modern turntables usually have an S-shaped or Straight tonearm. The most common type is the S-shaped, since it gives a better signal-to-noise ratio, and some premium turntables have devices with which to switch the tonearm from S-shaped to Straight, and vice versa. Two turntables that allow this are the Numark TTX-1 and the Numark TT-200. Why would someone want to switch to a Straight tonearm? Well, basically, it gives better performance when you are doing heavy scratching.
At the base of the tonearm is what looks to be a pretty complicated apparatus. This serves two purposes-- to route the tonearm's signal to the output of the turntable, and to provide a way with which to adjust the tonearm. There will usually be several controls to do this with. I will not go into too much detail here, as it is mostly already covered in How to balance a tonearm. There will usually be a height control. This sets how high the tonearm actually is off of the record. It is usually a smallish lever. Moving it up will make the tonearm go up, moving it down will make it go down. There will be a weight control. This is essentially a small weight that you can screw in or out, so that the balance on one side of the tonearm will match that of the other side. The last control that should be there is the anti-skate. This should be a big dial on the end of the tonearm. The anti-skate basically kind of tilts the tonearm, to prevent skipping and keep the stylus from moving side to side. If you intend on doing any backcueing or scratching, make sure to set the anti-skate to 0, or your stylus will break off.