The currency used in Budapest and all of Hungary is the Hungarian Forint. Despite being part of the European Union, Hungary does not yet use the Euro, though it's easily exchanged.
Currency conversion is readily accessible by the various exchange shops lined along Vaci Utca, at the airport, and at the train stations and most major shopping areas. While some people have recently reported seeing large differences in rates offered by various exchange offices, generally they hover around the same price.
ATMs are varied and frequent throughout Budapest, so using your bank card is not a problem so long as your bank is part of the Cirrus or Plus system. Just look at the back of your card for one of those logos or ask your bank before you leave. Most banks will charge you a fee for the transaction, so it generally makes sense to take out a larger amount of money instead of smaller amounts.
Credit cards are always accepted at major hotels, however credit cards in general are not nearly as accepted in Budapest as they are in North America for example. Some shops, when you use your credit card for payment, will ask to see some form of identification such as your passport. If your credit card disappears with a shop keeper for a longer period of time, you may want to monitor your next credit card statement to make sure there isn't any suspicious activity on it. That being said, credit card fraud isn't necessarily as large a problem as in most Western countries.