Tripod selection comes down to a few different compromises. What is more important to you, the cost, tripod weight, or the stability of the tripod? It is a set of trade offs and you have to choose what is most important to you, or decide that you won't get the best of all three.
- Stability/Capacity - Are you shooting with a small point and shoot, or a pro DSLR with a big telephoto lens? Look for maximum weight capacity specs and keep these in mind. You don't want to overload the unit and risk your camera hitting the ground if a bit of wind comes along.
- Leg Material - You can choose between carbon fiber, aluminum or some more specialty options such as (gorilla pod legs and other fibers).
Aluminum is usually the cheap option, they are cost effective, but not very light or great at dampening vibrations. Carbon fiber is typically the preferred material, as it is light, solid, and dampens vibrations.
- Price - Sure you can pick up a tripod for $15USD or so, but putting a $500 DSLR on top of that in on top of a windy rock, is not a good idea. A quality tripod will go a long way to capturing a great image.
- Height - How tall are you, and what is a comfortable height to shoot from? As suggested in the comments below, you want to think about what level you want to shot at, not necessarily your height. Although personally I think both are important :) Keep in mind that many tripods have separate leg and ball head sections, that can add height. Many have options for a center column extension, and some even have options to get down low to take macro shots.
- Size - You have to consider not only the fully extended height(above), but the size of the tripod and head when collapsed. This is important if you are hiking or traveling with the unit. Usually tripods with more sections are smaller, but also hold less weight.
- Leg Releases - This can be a personal preference thing, but each one of us usually likes one or the other. Some legs twist and some have levers. I would try each out yourself to see what you like.
- Head Configuration - This gets into a large subtopic. Basically you can get a tripod with a built in head, that is not user switchable, or a unit that is essentially separate, and allows you to use a different head as desired. It is for the most part desirable to have a user switchable head, but it usually costs more to have this option and buy the separate pieces. A few examples of heads include - Ball heads, still camera heads, video heads, shotgun style heads, gimbal style heads, geared heads, and panoramic style heads. Usually the inexpensive models have the still camera heads, and the professional lines tend towards the ball head style. Much more info about heads can be found in this previous answer - What should one consider when choosing a style of tripod head?
- Release - The release mechanism is quite important. Most use some type of quick release plate, the plate essentially gets secured to your camera during the days shoot, and allows you to quickly mount the camera to the tripod without screwing things in and out each time. Some lower end units or small specialty tripods will use a simple screw mount with no release plate. This works well if the tripod is very small and simple in design.
Other specialty features
This could explode into a huge topic, but I'll try to cover the basics.
- Macro photographers like to use "low angle adapters" or "inverted center columns" to get low to the subject such as in this question.
- Monopod for ultra mobility and size/weight considerations, or to assist in panning while using large lenses.
- Gorillapod or other segmented legged tripod for attaching to various objects.
- Beanbag objects(not technically a tripod, but useful for similar requirements)
- Lens collar which will provide better balance when mounting large heavy lenses(more of an accessory, but comes into play when considering tripods)
- Bubble levels
- Degree scale markings on the tripod head
- Center column attachment or hook for added weight(helps with stability)
Overall, you can see this is a very large topic, that takes some consideration on your part to decide what is important to you. If size and cost are extremely important, a Joby GorillaPod might be the best option for you. If stability and weight are the most important, you might want a Gitzo GT3541LS leg with a Gitzo GH2750QR head... too many choices :)