A variant is a version of a game that has branched off from the original and develops along its own, different, path. Angband has a great many variants. Nethack has typically had one active variant at a time. Features from variants are sometimes imported back into the original game as it develops.
Variant is a relative term and new variants can be spun off from older ones. Angband began as a Moria variant and Angband spawned many variants including Zangband, which in turn was the basis for various Zangband variants. Of these, ToME became quite prominent and in recent years has spawned the T-Engine, designed to produce games based on the same engine which might also be considered a form of variant.