The wise centaur who served as tutor to Achilles and Heracles was Chiron. Unlike many centaurs in Greek myth, who are portrayed as unruly and violent, Chiron was celebrated for wisdom, medicine, astronomy, prophecy, and teaching.
Chiron was said to be the son of Cronus and the Oceanid or nymph Philyra. He lived on Mount Pelion in Thessaly and became a mentor to numerous heroes. Peleus entrusted the young Achilles to him, and Chiron trained Achilles in hunting, combat, music, and healing. Ancient and later traditions also connect Chiron with Heracles, who learned from him and eventually caused his fatal wound accidentally.
Chiron’s immortality made his death especially tragic. A poisoned arrow belonging to Heracles wounded him, and because the injury could not heal, Chiron eventually surrendered his immortality. In one tradition, this allowed Zeus to free Prometheus from punishment.
Pholus and Nessus are also famous centaurs, but neither is the canonical tutor named in this question. Chiron’s name is sometimes written Cheiron or Kheiron.