The black market can be located on the east side Sanctuary (Hostel). The market is a place where you can purchase SDU upgrades. These SDU's will give you increased clips for all guns or either more spaces for your personal equipment bank.
To buy any of the SDU's you'll first need some Eridium which can be found via loot drops, treasure chests or completing missions. You'll now need a total of 20 Eridium to purchase 5 different things as the initial lot all cost 4 Eridium each. Below is a list of all the items for sale.