These tiny (1/50th of an inch) eight legged creatures resemble minute spiders. From a plants viewpoint they resemble aphids in that they feed by sucking the juice from plants and when present in sufficient numbers they can weaken, stunt or even kill infested plants (or parts of them). The first signs of damage are small light colored specks on the leaves, and in extreme cases these may become discolored and scorched and eventually die and fall off. There may also be downy webs (they produce these for protection) on the undersides of the leaves. In warm dry weather (or in a greenhouse) they can multiply rapidly, producing a new generation every couple of weeks.
There are many species of spider mite, but one of the most important is the Two-Spotted Spider mite (Tetranychus urticae) which attacks a wide range of plants, including beans, eggplant, raspberry, strawberry, fruit trees and more. Some overwinter as adults and some as eggs.
The most important control for spider mites is a healthy population of predatory insects, so do everything you can to encourage them (you can actually buy predatory mites for controlling them). Do this by providing them with habitat, planting lots of nectar producing plants (especially from the Daisy and Carrot families) and not killing them with pesticides. If you have an infestation that requires more direct action, try is washing them off of plants with a strong jet of water. If this fails you might try insecticidal soap, neem (neem seed oil works the best) or as a last resort, rotenone (but remember that they most often become a problem because indiscriminate use of pesticides kills off natural enemies).
Image: Louis Tedders, USDA Agricultural Research Service, Bugwood.org