You get this stuff called Eridium as a drop item. Which you can trade to a crazy guy in Sanctuary who runs the black market, and he sells SDU's for bank storage and backpack storage. +3 storage for backpack per SDU and +2 storage for bank per SDU. the bank initially starts with I think 6 slots, the backpack starts with 12. I have no idea if the bank SDU also affects claptraps secret stash(which is a shared stash between all your characters so you can easily swap items between them)
Shot through the knee and you're to blame! You give adventuring a bad name!