What is the name of the musical form where a theme is repeated with variations?

The musical form in which a theme is repeated with variations is called theme and variations.

In this form, a composer presents a recognizable theme and then restates it several times, changing features such as rhythm, harmony, melody, texture, register, or instrumental color. The theme usually remains identifiable even as each variation gives it a different character. A theme may be a newly written melody or a borrowed tune.

Theme-and-variations movements became especially prominent in the Classical period, although the technique is much older. Beethoven's Diabelli Variations, Brahms's Variations on a Theme by Haydn, and Elgar's Enigma Variations are well-known examples. The form can stand alone as a complete piece or appear as one movement within a sonata, symphony, or chamber work.

A fugue develops overlapping entries of a subject, while a rondo repeatedly returns to a principal section between contrasting episodes. Sonata form organizes contrasting thematic areas and a development section, so none of those terms specifically means repeated transformations of one theme.