Currency, or money (we'll use the terms interchangeably for the purposes of this discussion), can be defined as a unit of purchasing power. It is a medium of exchange, a substitute for goods or services. It doesn't have to be the coins or bills with which you're probably most familiar. In fact, through the ages, everything from large stone wheels, knives, slabs of salt, and even human beings have been used as money. Anything that people agree represents value is currency.
For example, if you have one barrel of wheat, and you want a cow, without currency you have to find someone who not only has a cow, but also wants a barrel of wheat and will agree to the trade.
Let's say your neighbor fits the bill -- he has a cow and wants a barrel of wheat. What if a barrel of wheat isn't worth an entire cow? Your neighbor can't exactly make change by giving you part of a cow.
Now, if you live in a place where round, stamped coins are widely considered to have a certain value and can be exchanged for other things, then you just have to find someone who needs wheat. That person will take the wheat in exchange for an agreed-upon amount of coins, which you can later use to buy a cow from someone else.
This is not to mention the fact that carrying a handful of coins is much easier than lugging around a barrel of grain or a cow with you whenever you need to make a trade.