Caraxes is the dragon of Prince Daemon Targaryen during the Dance of the Dragons, having previously been ridden by Prince Aemon, one of Jaehaerys's heirs. Throughout the civil war, Caraxes and Daemon fight many battles, supporting the claim of Daemon's wife, Rhaeynra. In the end, Caraxes and his rider die in battle. Daemon challenges Aemond and his dragon, Vhagar, to a great battle, which ends with all four dead.
Daemon is one of the main characters in "House of the Dragon," so it's all but certain that we'll spend some time getting to know Caraxes on the show.