The Great Vault is located in Oribos, in the room just counterclockwise from the Inn. In addition to the usual bank and Guild bank access, there is an upgrade to the "box of disappointment" from the previous expansions. Now, instead of just giving you a bonus piece of loot based on mythics/pvp, it gives you a bonus piece from mythics/pvp/ or raids.
Each week, you can select only 1 piece from the choices presented to you. You have more choices based on doing more things--if you kill 3 raid bosses you unlock one choice (one box), if you do a mythic keystone, you open another, if you clear the raid you'll open up a total of 3 choices, etc. Normally to see what your choices are for the week, you have to go to the vault to check it, but there is a magic incantation:
/run LoadAddOn("Blizzard_WeeklyRewards"); WeeklyRewardsFrame:Show()
that will show it from anywhere.
If none of the rewards are interesting, you can always choose the stygia tokens. If you did a mythic plus in the previous week, there will also be a new key for you.
A couple of oddities: the item level of the rewards is based on what you do to fill in the box. If you kill 3 raid bosses heroic, and then 3 more on normal, the first raid box will have a 213 item as its choice, while the other will be lower. If you kill the Stone Legion Generals or Sire Denathrius, the box will have slightly higher ilvl pieces (207 for normal). For the raids, the ilvl of the rewards is based on the content you do; it's of the same level as what you killed to open the box up.
For the mythic +'s line, it's a bit more complicated. If you do any mythic plus, the first box will be open with a reward from about 3 difficulties higher than what you did. If you do enough mythic pluses to open the next choice box, they look at the lowest mythic plus that you ran to get that many of them, and give you 3 above that.
One thing I just discovered--if you forget to pick up your vault item for a week, the next week when you show up there will be a note at the bottom: "These results were generated over a week ago." When you select that item, close and reopen the vault, it will show you the next week's choices. It doesn't lose anything.