I see tripods that are metal, carbon fiber, basalt, and even wood. What are the features and/or drawbacks of each of these? What factors should be considered when choosing what material they want in a new tripod?
Basalt refers to the source of carbon fiber. So, I would bunch these two into one.
Carbon fiber and basalt have an excellent weight to rigidity ratio. In other words, it gives the same strength for less weight. The process of creating carbon fiber is complex and has to be woven into tubes using several layers (5-8). This gives it its main disadvantage which is cost. The second disadvantage is that in extremely cold temperature, carbon fiber shatters.
Metal is relatively cheap to produce. Very strong tripods can be made of metal but they will be quite heavy. There are variations among metals. These include aluminium (lighter) and magnesium (stronger).
There are also hybrids where some tubes are carbon fiber and some are metal and others where tubes are made of both materials. These are designed to offer a compromise between the two classes of materials.
(The main differences between basalt fiber and carbon fiber is basalt is cheaper to produce, thus the tripods are less expensive, but not as strong as carbon so the tripods created using basalt are ever so slightly heaver and can hold ever so slightly less. Though some of those tradeoffs are surely exacerbated by the manufacturer by having the overall build quality of the CF tripod be higher in order to justify the greater expense of the carbon fiber tripod)
As far as vibration dampening and rigidity in general, the more material used in the construction the more rigid and dampening it'll be, relatively. Generally a CF tripod will be more rigid/dampening than an aluminium tripod but a 100lb solid block of aluminium with a head mounted on it will win that contest.