The word 'variation' in music refers to the process of repeating musical material in an altered form. The alteration can involve either melodic, rhythmic, harmonic, or timbric aspects. In a broader sense, the use of variations as a way of generating musical discourse has resulted in a number of forms or genres. The passacaglia and chaconne, for example, are variation forms.
The most popular variation form is, however, the 'theme and variations'. A fundamental musical idea is stated (with or without an introduction) and then recurs a number of times, altered in its melodic shape, instrumentation, meter, rhythm, accompaniment, or some other aspect. Usually the variations will have the same structure and length as the original theme (which may be considered a legacy from the application of the variation technique to court dances).
The first theme/variations works emerged in the 16th century, under the form of divisions (in which a melody was stated in long notes, and then subsequently divided in a progressive fashion, thus demanding some level of dexterity of the player). The basic idea of stating the variations in a progressive fashion has been a constant through the history of the Variation forms, arguably because it creates a sense of form and musical direction.