Gubraithian Fire (or Everlasting Fire) is fire enchanted to burn forever. The enchantment seems to be an advanced-level of magic, as not many wizards are capable of doing it. Because of that, they are highly prized and presumably kept as exotic treasures. Rubeus Hagrid and Madame Maxime gave a branch burning of this particular Magic-Fire to the Gurg of the Giants as a gift and proof of good faith.
Albus Dumbledore is one of the known wizards who can conjure it and was, in fact, the wizard who provided the branch to be brought to the Gurg. Professor Flitwick had mentioned it in Charms "at least twice" before then, according to Hermione Granger.
The name "Gubraithian" is evidently taken from the Scottish Gaelic phrase "gu brath", which means "forever".
- Harry Potter and the Order of the Phoenix (Mentioned only)