The Vault is a "secret" area in Diablo 3 where there are lots of goblins, and a goblin boss at the end. He drops a guaranteed two legendary items, as well as a unique legendary gem. This is a good place for farming gold, gems and Blood Shards, and there are currently a few different ways to get to The Vault:
1) Use a Puzzle Ring in Kanai's Cube. This will destroy the ring, but open a portal to The Vault. The Puzzle Ring is obtained from random drops like the rest of the legendary items in the game, but gambling for rings at Kadala will usually give you one pretty easily.
2) When playing in Adventure Mode, any goblin you kill will have a low chance to open up a portal to The Vault (excluding Rainbow Goblin). You can then enter the portal and clear the Vault, and get teleported to the area you killed the goblin in upon finishing.
After clearing the final boss in The Vault the portal will close after 90 seconds, and you are free to open a new portal if you got another Puzzle Ring. This lets people group up and clear several instances of The Vault very fast, which is a nice way to get gems and gold at the start of a new season.