Proof of Work
Proof of Work is system of requiring a node to generate hash with a certain amount of zeros at the beginning. It can take many attempts to generate a hash with the sufficient amount of zeros, so requiring such a hash can be resource intensive and therefore can be considered a "proof of work" performed.
Proof of Work systems have been utilized by email protocol to cut back on nodes sending many emails out to multiple addresses, or in other words sending spam. By requiring the sender to provide a proof of work hash, sending out a large number of emails would be resource intensive and may not be worth the cost for potential spammers.
The Bitcoin network and many cryptographic currencies use Proof of Work to confirm transactions and add new transactions and coins to the blockchain, this is known as Proof of Work mining. Proof of Work mining gives coin rewards to the miner that successfully creates a hash (SHA-256 for Bitcoin or Scrypt for HoboNickels) with at least the required amount of zeros.
Proof of Work mining can be performed with either CPU's (via Minerd or similar programs) or GPU's (through CG Miner or similar programs). Most serious miners use GPU's because they process the algorithms much more efficiently and at higher hash rates than CPU's.
Proof of Work mining can be done by one worker, known as solomining, or by connecting to a mining pool that connects miners to a group of devices mining and distributes blocks according to worker share.