A currency is a
brand of
money. Today, these are usually issued by large
policital entities - most often
nation states, but occasionally also
unions (e.g, the
euro, being issued by the
European Union.)
Trading in currencies is called Foreign Exchange or FOREX. It is based on the variations of price between various currencies, and involves staggering amounts of money. How different currencies are priced against each other varies depending on the economic climate and how the international agreements about how to use currencies are at the time. Historically, there has been a number of different schemes.
For centuries, it used to be common for currencies to be backed in gold, letting the person possessing currency exchange it for gold. This became more or less universal in 1879, and lasted until the outbreak of World War I in 1914. From 1914 to 1945, currencies sort of floated depending on the whims of the issuers. Then came the Bretton Woods agreement in 1945, adding back a gold tie, but allowing countries to adjust their par rate to gold on an uniliteral basis if the IMF agreed. The next change was in 1950 - where the gold standard was replaced with dollar standard, with the USA having a fairly passive role, just trying to keep prices of tradable goods stable and allowing the access to the US capital markets open for everybody. This policy was called the Fixed-Rate Dollar Standard, and lasted until 1970, and was followed by a brief period of instability. In 1972, it was replaced with the Floating-Rate Dollar Standard, in which countries tried to keep their currency reasonably stable against the dollar in the short term, but did not commit to a stable rate of exchange over time. The US also started pursuing a more active role, not trying to keep prices of tradable goals stable. This lasted until 1984, and in 1985 was replaced with an attempt at keeping the rate between dollar and DM and the rate between dollar and yen reasonably stable as long as economic fundamentals didn't change drastically.
From 1972 and onwards, most currency rates have been floating. This means that the currency isn't based in anything but trust in the issuer. There is no fixed exchange for anything else; everything depends on the present rates in the foreign exchange market.
The main differences between a currency and a stock is that the currency is intended for use in generic barter, is often backed by a larger entity than the stock, does not pay a dividend, and does not give voting rights. A currency also usually has a negative effective rate of return due to inflation.