Rapture has its own currency, the Rapture Dollar (R$), with Ryan's unsmiling face on the front.
Dollars are everywhere, lying around in clips, scrounged off dead aggressors or (muy mucho dinero!) Big Daddies, and stored in safes, lockboxes and cash registers.
You spend money in vending machines, in health stations, and in bot shutdown switches. You can also spend money in order to save you the trouble of cracking safes or security, or to lower the price and raise availability of goods in vending machines.
Your wallet is limited to R$500. How much you have in that wallet is given on the left in the menu screen.
Spending money to crack a vendor or health station makes sense only if you intend to use that machine repeatedly. If you are low on money, solve the hacking puzzles for the machine or use an auto-hack tool.
Although lockboxes can appear by themselves, they usually pop up in place of a fallen aggressor after a minute or two in order to save on graphics processing in the game — much like a demon dissolves into smoke in Doom 3.
Big Daddies have lots of money, probably because volunteers were paid all that in order to lure them into becoming Big Daddies. Once the process was complete, ironically, they got to keep their money even though they were no longer able to spend it or even to understand what it is. You win its money as a fringe benefit of destroying it to free/harvest the Little Sister it guarded.