There are four genera of interest in this family, Acropora, Anacropora, Astreopora and Montipora. Those in the Acropora genus are quite diverse and popular and having different forms of growth, e.g., branching, cluster, bushy, and finger-like. In fact, these photosynthetic corals are the most abundant in the entire world. Some are difficult to ship and acclimate, but once established most exhibit rapid growth. These small polyp stony (sps) corals (sometimes referred to as "small polyp scleractinian" corals) mostly require intense light (with metal halides being preferable), excellent water flow and can easily be propagated by snapping off branches and relocating in areas of favorable conditions. Unfortunately, their common names leave much to be desired, as they are often related to shape or color, so common names are questionable. Many are shown here, and General Husbandry has been limited because the desire here is to show the species, and not be repetitive as to their needs, as most simply require excellent lighting and moderate to swift water movement.
As to those in the Montipora genus, they are another very large family of photosynthetic corals similar to Acropora with many different shapes/forms and extremely small polyps. They are also quite popular with reef aquarium hobbyists and the same direction will be taken with their husbandry. But I should add, I've found my digitata species requiring very excellent lighting and strong water movement to retain their individual colors, e.g., green, orange and blue.
Those in the Anacropora genus are closely related to Montipora and until recently were not seen in the trade. They are a slow growing species, rarely seen in the trade, and are best kept low in the aquarium, as they often come from turbid conditions/muddy bottom areas.
Astreopora species are more mound-shaped or encrusting species, and also rarely seen in the trade. In fact at one time they were a species listed in the Turbinaria genus. Mostly, their polyps extend at night, and this may be a genus of corals best left in the wild.