My daughter has been taking ballet since she was five, and it’s safe to say she is a very enthusiastic fan of the art. 😉  A few weeks ago she asked me to look up where the word comes from, so of course, I obliged.
Ballet comes to English from (shocker) French. Though interestingly, the original French word is actually the feminine ballette. It’s been in English since the 1660s, meaning “a theatrical, costumed dance telling a story.” The French word is, in turn, from the Italian balletto, which is a diminutive of ballo, meaning “to dance.” The Italian, of course, come from Latin (ballare) and the Latin in turn from the Greek ballezein, all of the same meaning.
What I don’t think I ever paused to realize is that this is also where ball, as in, “a large dance party,” comes from (duh, right?). Perhaps simply because we pronounce ballet and ball differently, my brain just hadn’t made that connection. But OF COURSE!