The character introduced in “The Empty Child” who later becomes a companion is Jack Harkness.
Jack first appears during the 2005 two-part story “The Empty Child” and “The Doctor Dances,” set during the London Blitz in 1941. He initially poses as an American Royal Air Force volunteer, but is actually a time-travelling con man and former Time Agent from the 51st century.
After helping cause a deadly plague, Jack helps the Ninth Doctor and Rose Tyler stop it. He then joins them aboard the TARDIS, becoming a recurring ally and companion. His later story becomes especially significant because he is made immortal after being killed by a Dalek.
A common mix-up is to identify Jack with one of the companions from the modern series, such as Martha Jones or Donna Noble. Jack is different: he begins as a guest character in the Ninth Doctor’s era and later becomes central to the spin-off series Torchwood.