I'm not sure I can do this subject justice, it has a lot of depth.

During the 18th century there were several types of currency in use heavily in the colonies. Gold and silver coins, which were typically Portugese or Spanish (such as the Spanish dollar). And paper money including not only British pounds but also money, pounds, printed by each colony (e.g. Virginia pounds, New York pounds, etc.) Colonial pounds each had their own exchange rates and since, unlike the British pound, they didn't have a fixed exchange rate with precious metals their value fluctuated and they often experienced significant inflation. Several acts of British parliament in the 1750s and '60s limited the use of colonial printed money, disallowing their use for anything other than paying taxes, essentially.

During the revolutionary war the Continental Congress began issuing paper money, denominated in dollars similar to Spanish money. The colonies began printing paper money again (denominated in either pounds or dollars) as well to pay for the war effort. Note that this paper money was not terribly standardized. It's almost better to imagine these notes as personal checks from the colonies that were circulated amongst colonists as money. There was a large diversity of face values on such currency, ranging from a handful of pence to seemingly random amounts of dollars or pounds (e.g. 2 pounds, 3 pounds, 8 dollars, 5 shillings and 5 pence, and so on). Moreover, banks themselves printed their own notes which were in circulation.

After the war the monetary state of the US was a mess. Continental currency suffered from heavy inflation and was also greatly impacted by successful British attempts at counterfeiting during the war. There were many different types of money floating around the colonies at the time (continental dollars, British and colonial pounds, Spanish and Portugese dollars, bank notes, etc.) Even after the constitution was ratified this was still a problem, though the institution of US treasury bonds helped stabilize the overall situation somewhat.

The Bank of North America was established as a private/public partnership to help deal with the problem, and it issued notes in the post-revolutionary period (and that role was then taken up by The First Bank of the United States shortly thereafter but then it stopped operation in the early 1800s). Also, a clause was added to the constitution to deny individual states the ability to print their own money. Moreover, in 1792 Congress established a fixed exchange rate for US dollars to a weight of silver. Despite all this there was still a substantial diversity of money being used in the US throughout the 19th century. The US mint issued coinage (in dollars) but this merely added to the other coinage and paper money being used in the US.

It was not until the mid 19th century before foreign currency in the US became less common. And it was not until an act of congress in 1857 which outlawed use of foreign coinage as legal tender before the US ended up with the more or less monolithic monetary system we have today.