Proof of work
A proof of work is a piece of data which was difficult (costly, time-consuming) to produce so as to satisfy certain requirements. It must be trivial to check whether data satisfies said requirements. Producing a proof of work can be a random process with low probability, so that a lot of trial and error is required on average before a valid proof of work is generated.
One application of this idea is a proposed method for preventing email spam, requiring a proof of work on the email's contents (including the To address), on every email. Legitimate emails will be able to do the work to generate the proof easily (not much work is required for a single email), but mass spam emailers will have difficulty generating the required proofs (which would require huge computational resources).
Proofs of work are used in Bitcoin for block generation. Proofs of work that are tied to the data of each block are required for the blocks to be accepted. The difficulty of this work is adjusted so as to limit the rate at which new blocks can be generated by the network to one every 10 minutes. Due to the very low probability of successful generation, makes it unpredictable which worker computer in the network will be able to generate the next block.
For a block to be valid it must hash to a value less than the current target; this means that each block indicates that work has been done generating it. Each block contains the hash of the preceeding block, thus each block has a chain of blocks that together contain a large amount of work. Changing a block (which can only be done by making a new block containing the same predecessor) requires regenerating all successors and redoing the work they contain. This protects the block chain from tampering.
Let's say the base string that we are going to do work on is "Hello, world!". Our target is to find a variation of it that SHA-256 hashes to a value beginning with '000'. We vary the string by adding a integer value to the end called a nonce and incrementing it each time. Finding a match for "Hello, world!" takes us 4251 tries (but happens to have zeroes in the first four digits):
"Hello, world!0" => 1312af178c253f84028d480a6adc1e25e81caa44c749ec81976192e2ec934c64 "Hello, world!1" => e9afc424b79e4f6ab42d99c81156d3a17228d6e1eef4139be78e948a9332a7d8 "Hello, world!2" => ae37343a357a8297591625e7134cbea22f5928be8ca2a32aa475cf05fd4266b7 ... "Hello, world!4248" => 6e110d98b388e77e9c6f042ac6b497cec46660deef75a55ebc7cfdf65cc0b965 "Hello, world!4249" => c004190b822f1669cac8dc37e761cb73652e7832fb814565702245cf26ebb9e6 "Hello, world!4250" => 0000c3af42fc31103f1fdc0151fa747ff87349a4714df7cc52ea464e12dcd4e9
4251 hashes on a modern computer is not very much work (most computers can achieve at least 4 million hashes per second). Bitcoin automatically varies the difficulty (and thus the amount of work required to generate a block) to keep a roughly constant rate of block generation. The probability of a single hash succeeding can be found here.
In Bitcoin things are a bit more complex, especially since the header contains the Merkle tree which depends on the included transactions. This includes the generation transaction, a transaction "out of nowhere" to our own address, which in addition to providing the miner with incentive to do the work, also ensures that every miner hashes a unique data set.