A digital currency relying on the principles of cryptography
The Wiki describes it well; a cryptocurrency (such as Bitcoin, Litecoin, Feathercoin, etc.) is a peer-to-peer, decentralized, digital currency whose implementation relies on the principles of cryptography to validate the transactions and generation of the currency itself.
Let's look at the first sentence again and break it down word for word:
A cryptocurrency is a peer-to-peer, decentralized, digital currency whose implementation relies on the principles of cryptography to validate the transactions and generation of the currency itself.
Peer-to-peer refers to each node on the network being able to connect to any other node and act as both a server and a client sharing updates to the blockchain (public ledger) without the need for a central server.
Decentralized refers to the peer-to-peer architecture of the network. Because each node can act as both a server and a client, there is no need for a central server or centralized authority, thus effectively solving the double spending problem.
Digital currency, among its various names, is electronic money that acts as alternative currency in the digital realm (though offline adoption is increasing as well). Currently, alternative digital currencies are not produced by government-endorsed central banks nor necessarily backed by national currency.
Cryptography relies on public and private keys for security. With cryptocurrencies every transaction has to be signed by a private key for security. Due to the strong security and well understood principles of cryptography, counterfeiting digital currencies is virtually impossible.