Many US banks have foreign branches, and many foreign banks accept deposits of US dollars. These types of accounts were first established in Europe and thus are denominated in what are known as eurodollars. A company that is based in the US can use a local bank for financial transactions of their foreign offices by depositing (or more likely wire transferring) US dollars and writing checks from that bank in US dollars instead of local currency.
Once the US dollars are deposited in the overseas bank, that bank can then lend them out without the restrictions of the Federal Reserve and its reserve rate. This lack of regulation has created a very large market in Eurodollars well beyond its initial scope. Facing no regulations or restrictions beyond the deposits they receive, banks could theoretically face a liquidity crisis if depositers decide to move their money back to domestic banks. The lender of last resort in case the bank does not have enough dollars is the Federal Reserve, but most banks have standing agreements with (non-Fed) US banks to prevent such a crisis.
One important concept with these deposits is that the currency never actually leaves the United States; instead, the liability of the deposit at the domestic bank is transferred to the foreign bank. Even if that liability is lent out, the actual money backing the liability remains on deposit in the US bank.
Information from http://wfhummel.cnchost.com/eurodollars.html and http://www.investopedia.com/university/moneymarket/moneymarket6.asp