A convenient medium
Paper money was invented by the Chinese in the 7th century as way of simplifying large monetary transactions; it is a lot easier to handle than thousands of copper coins. The Chinese used paper money for centuries before Europeans began to use it during the 17th century.
The first Europeans were the Swedes, who developed paper money for reasons similar to those of the Chinese - the Swedes had an abundance of copper coinage that was difficult to use due to its weight and bulk.
By the end of the 18th century, paper currency was in use throughout most of Europe and its colonies.
Paper money developed in two forms: Drafts, which are receipts for value held on account; and Bills, which were issued with a promise to convert to "real" money, i.e. coins with value based on their metallic content.
The value of paper money before the middle of the 20th century was dependent on what it could be exchanged for - paper money had no intrinsic value of its own.
Thus most paper currency specified that it was exchangeable at a location such as the Treasurer's office, or a specific bank, for a specified amount of silver or gold coinage.
Modern paper money is rarely backed in this way - most world paper money is now backed simply by the issuing government's promise to accept it in payment.