Black magic is a branch of magic most commonly found in Japanese role-playing games. It deals mainly with offensive elemental damage-dealing spells and is the opposite of the healing branch known as white magic.
In the Final Fantasy games, black magic most commonly encompasses these basic spells - fire, ice, water, thunder, earth, and gravity. Each are given more powerful versions and denoted with a varying suffix - as an example, the fire spell's progressively more powerful versions are almost always called fira and firaga, and deal more damage and/or target more enemies depending on the entry in the series.
Most other JRPGs handle black magic in a similar manner. Sometimes an extra branch is included for the sake of grouping together time-altering spells, but they generally fit under one of the two main headings.
Western role-playing games usually do not group spells the same way - they usually sort them by element type, effect, or the level required to cast them.