Vault 713 is the “top security” vault underneath Gringotts Bank where Dumbledore kept the Philosopher’s Stone (PS5).
- Harry and Hagrid visited the Vault on Harry's eleventh birthday, July 31, 1991. Hagrid showed the Goblins a note from Albus Dumbledore so he could gain access to the Vault. Harry expected to see jewels and gold inside, but there was only the "grubby little package" concealing what he later realized was the Philosopher's Stone (PS5, PS9).
- The Vault had no keyhole, but opened when Griphook stroked the door with his finger. He told Harry that if anyone except a Goblin tried to open the door, he would be sucked inside and trapped, and they only checked for prisoners about every ten years (PS5).
- Vault 713 was broken into the same day by Professor Quirrell, who was trying to steal the Stone for his master, Lord Voldemort. Griphook told the Daily Prophet the Goblins knew nothing had been stolen because the contents were removed earlier, "But we're not telling you what was in there, so keep your noses out if you know what's good for you" (PS8).
- The inability of Quirrell to steal the Stone from Vault 713 led to Voldemort possessing Quirrell, hiding himself within the turban on his head (PS17).
The number of Vault 713 is the reverse of Harry's birthday 7-31 (and J.K. Rowling's birthday too). Of course they visited the Vault on Harry's birthday as well.
The Vault number may also be seen as magical, combining lucky number 7 with unlucky number 13.
It's a good guess that the Vault belonged to Albus Dumbledore since his permission gave Hagrid access to it, but it could also have belonged to Hogwarts School or Nicolas Flamel himself, since he was a good friend of Dumbledore's.
It is unclear how Professor Quirrell was able to avoid being sucked inside Vault 713 when he attempted to steal the Philosopher's Stone. Some sort of alarm must have alerted the Goblins because they knew of the break-in, however they also knew the culprit got away. Voldemort must have told Quirrell a cunning way to escape Gringotts undetected.