The Mongolian unit of currency is the togrog (T), which comes in notes of T5, T10, T20, T50, T100, T500, T1000, T5000, T10.000 and  T20000 (T1 notes are basically souvenirs). We used to use also T50 and T100 coins. The highest value note is worth around US$9 so when changing a lot of cash you'll be given a stack of machine-collated bills.
Banks and exchange offices in Ulaanbaatar will change money with relative efficiency. Those in provincial centres have also gotten better, though it's still a good idea to change the bulk of your travel expenditure before leaving the capital.
Other forms of currency except US dollars aren't usually accepted, although the euro is probably second best. Cash offers the best exchange rates and you won't be paying any commission charge, but for security purposes you can also use travellers cheques and debit cards.
Moneychangers who hang around the markets may or may not be legal. They offer the best rates for US dollars and are usually safe, but the risks are obvious. Remember to change all your togrog when leaving the country as its worthless elsewhere.

The Trade and Development Bank has plonked down ATMs at a few key locations in Ulaanbaatar, Darkhan and Erdenet. These ATMs accept Visa cards and work most of the time, allowing you to withdraw up to T400.000 per day. Because most of the Golomt Bank branches are open 24 hours, they don't have ATM machines. Ordinary ATM cards issued from your bank at home probably won't work; try to get a 'Debit' card linked to your bank account. It should be associated with a credit card company.