A dominant, in a BDSM context, is a person who exerts control over the will of another person: the submissive. The dominant may or may not inflict physical punishment, but the submissive always takes direction and instruction from the dominant.
Often the terms dom (male) or domme (female) are used to refer to such a person. A pro-domme is a domme who is a professional sex worker. The term dominatrix can mean either domme or pro-domme. Often, the d in these terms is capitalized (i.e. Dom) to signify the dominace of that role over the (lower case) sub.
Some people have a more or less fixed orientation as a dominant, while others assume the dominant role for a limited time, perhaps only a single scene. A person who is at some times (or with some people) a submissive, and at other times a dominant, is often called a switch.