Corals are beautiful, fragile organisms. They are also expensive to purchase and difficult to care for. Given the problems that coral reef ecosystems cause their natural environments, such as pollution and overfishing, owning and caring for them has become more than just a hobby.
Reefs are usually extensive open water systems with plenty of room for everyone, so you need to be prepared for what can happen when you put these seemingly passive, docile creatures in a closed space.
Believe it or not, corals fight. If two species are vying for space, one will sometimes send out its tentacles along the front line, the boundary between the two, and sting the neighboring invader or attacker. Sometimes a dead-zone exists between two adjoining coral heads, simply because when either coral species tries to settle in that space, a little war wages between the enemies.
In some cases, the corals will emit certain chemicals that are toxic to other coral species. Although certainly one can find examples of coral wars along a natural reef, generally, in nature, there's plenty of space to accommodate these incompatibilities. You just have to make sure not to create them in your tank.
The following list organizes corals (and their cousins, the anemones) from most aggressive to least aggressive. To know which ones you can keep together, look up each coral you are considering to purchase. Although you cannot avoid one species being dominant in a natural system, if chosen correctly the corals will live together in harmony.
Anemones can move around, but would rather just stay where they are. If you own only two species, chances are you can position them so that they won't be in contact. But once you begin filling the tank, you must be aware of proximity. Without even knowing it, you may be determining who's the boss. Conditions in your tank may favor a particular species, and eliminate the upper hand the other coral may have had.
Another "weaker" species may be very happy where you placed it, may be getting better light or more food. If you keep two species of corals in the same tank and aren't sure how they will respond to each other, make sure to keep them separated, they won't move once they're settled like its cousin the anemone can. Most won't mind staying in the same tank, unless space becomes an issue - especially if they are close together on the list below, where it may not be immediately clear who is dominant and a hierarchy battle can ensue:
Coral Species Dominance: From Most to Least Aggressive
Elegance Coral (Catalaphyllia jardinei, aka Toothed Coral)
Hammer Coral (Euphyllia ancora, E. fimbriata, aka Torch Coral)
Other Euphyllia species
Bubble Coral (Plerogyra sinuosa)
Grape Coral (Physosyra lichtensteinei, aka Small Bubble Coral)
Mushroom Coral (Fungia actinoformis)
Flower Pot Coral (Goniopora sp.)
Telia Anemone (Telia sp. aka Strawberry Anemone or Colony Anemone)
Open Brain Coral (Trachyphyllia geofroyi)
Cup Coral (Tubinaria peltata)
Moon Coral (Galaxea fascicalaris)
Closed Brain Coral (Favia sp. aka Dead Brain coral)
Star Polyps (Pachclavalaria sp.)
Leather Coral (Sarcophyton sp.)
Tree Coral (Sinularia sp. aka Fire Coral)
Corgoniana (Gorgonacea sp.)
Waving Hand Coral (Anthelia sp.)
Xenia (Xenia sp.)
Giant Mushroom Coral (Rhodactis sp. aka Elephant Ear Coral)
Sea Mat Anemones (Zooanthus sp. aka Sea Mat Rock
Ricordia Anemones (Ricordia sp. aka Sea Mat Rock)
Mushroom Anemones (Actinodiscus sp. aka Mushroom Polyps)