In photography, a tripod is one of your most important tools. It normally has three legs, usually telescoping.
One talks about studio tripods and field tripods, but anyway when you are in a studio you don't really want a tripod: you want a stand, a big columnar contraption from which you can hang cameras and flashes.
A tripod mounted on a dolly is a cheap substitute.

Anyway, tripods have two components, often made by different makers: the legset and the tripod head (of which there are different kinds).
One of the marks of a crufty tripod is precisely that the tripod head cannot be changed.

What should one look for in a tripod system ? Chiefly stability, which is more or less proportional to weight. Secondly, it must be practical to set up, tear down and carry around.
Opinions differ as to materials: carbon fiber is supposed to be really good and light, but it certainly is very expensive. Other things, like a good quick release system are really attributes of the tripod head.

Effects on your photography

A tripod is indispensable for certain kinds of cameras, like view cameras, but how will it affect your average small format photographer ?

  • Getting the good light: the good light for landscape photography, that's to say around daybreak and sunset is not abundant. A tripod will allow you to shoot at the speed you like, which means that you get to choose the depth of field (for example).
  • Night photography: it usually implies exposures that are measured in minutes. It can be attempted without a tripod, but it is inconvenient.
  • Macro or still-life: without a tripod, the minute camera movements and adjustments become hell. The genre is already painful enough by itself, but with a tripod it becomes bearable.
  • Portraits: once you have your camera firmly mounted on a tripod, you can actually take your eye off the viewfinder, make eye contact with your subject, and chat with them. Some people just need to look you in the eyes.
  • Nature photography: inconvenient as this may be, there are some nature pictures that just can't be taken handheld. Case in point: shooting birds at dusk with a 600 mm telephoto lens. Try that handheld.

How do I get one ?

Pick a good brand, like Gitzo, Bogen/Manfrotto. Expect to spend between $100 to $200 (it is less than most lenses !). Don't buy a model heavier than what you can carry with ease, and for quite a while.
Touch it, and ask yourself if it feels right. As with most photographic equipment, the rapport between you and the machine must be visceral. The dumb piece of metal must become first like an old buddy, and then like an extension of yourself.

You should not think "Now I will open the legs, now I will set the head". You should think "I want this image, the camera must be here", and your tripod should be a consequence.