Eurocurrency is money in the form of bank deposits of a currency outside the country that issued the currency.
The use of the prefix "euro" is somewhat misleading. It is used for historical reasons. Eurocurrency deposits may be of any currency in any country.
The most common currency deposited as eurocurrency is the US dollar, and the term eurodollar is often used to refer to dollar deposits.
One advantage of eurocurrency from a bank's point of view is that it allows a bank to operate outside the regulation of the country issuing the currency in question. One particular advantage is that most countries do impose a reserve requirement on foreign currency operations: a bank can therefore lend a higher proportion of its eurocurrency deposits, improving its interest margin.
It also makes sense for a bank that has customers who wish to hold or borrow foreign currency the convenience of doing so in their own country.
Sufficiently large borrowers often issue eurobonds instead of borrowing from banks' eurocurrency operations.