Proof of work
A proof of work is the verifiable result that can only be obtained through a given work. Proofs of work are hard to obtain (i.e. require work), but trivial to check. Proofs of work can be applied to information, you can prove that you did work on a particular number.
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).
This concept is used in bitcoin for block generation. 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 a predecessor block, thus each block has a [chain|block 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.
Let's say our base work is "Hello world" and our work is to find a hash starting with "000". We are going to increase an integer value (nonce) hashed alongside "Hello world", until we reach our goal.
Hello world ! 0 : 3f6fc92516327a1cc4d3dca5ab2b27aeedf2d459a77fa06fd3c6b19fb609106a Hello world ! 1 : b5690c48c2d0a09481186aaa99e4e090901ff2ac4d572e6706dfd30eefc22a27 Hello world ! 2 : 5b6fd9c27fcb54ca23404d9428f081b7c9280ba6370e33a6a20b16f40ce76320 Hello world ! 3 : 9c5d769416aa0ca894abf22bd17bd30fbb6959291423ae1903a9f86a1fe7ce78 Hello world ! 4 : 4efc65df7933e4f5cc21947c61d5cc6bd11d644794bfa210603b0547c4b1cc3e Hello world ! 5 : 441b15b67d791620cd50ea537144e3115422e33bdb1b1b9b110d3265f7a9199b Hello world ! 6 : d368331386f0cf773ad53910fefcef4bdceeb526e408d3fbc9408d6f6e481ca4 Hello world ! 7 : 013cc9722f38d2eb6186b75e2e7cbe6e7818e0612a2774d4400416b17ae03b87 Hello world ! 8 : 3a92631799b478c3bcc554df8401b09900fbdb58cc0e58efe711cc475ee097b3 Hello world ! 9 : 66658881696164fcb04f32ec505bb5e515000a85baf691beb63fc9d3f4d0fee2 .... Hello world ! 88 : 80d009db72c6ad35241bb3dbac77cbe177c6a803fe67527c159dbfaf2cbf9f5c Hello world ! 89 : a5b1e789f691f9793f8a84f8ebae3d8e28d49cbe0eeea2da621cd409e3bdee2b Hello world ! 90 : 4eba5b2459caac3d9ff3b787aaa5cac481aaa4a0232fbbe02a8ee4d1101c2ca2 Hello world ! 91 : c811722c68b53614d58d37dcad9d540c2bce9f85b5ccae94424ff4716eea1765 Hello world ! 92 : e30c716fccda22f394a8e80a2670b97968b5416b8b39e2061a7b7d1a9f41e0a9 Hello world ! 93 : 965425c39d4e24c532721d7f7b77a00b31b0c0d0e316d46240c4e6bec9c09f65 Hello world ! 94 : 7090a0e5d88cff635e42ea33fcd6091a058e9cdd58ab8cd5c21c1c70421e35c6 Hello world ! 95 : b74f3b2cf1061895f880a99d1d0249a8cedf223d3ed061150548aa6212c88d43 Hello world ! 96 : 447ca2fa886965af084808d22116edde4383cbaa16fd1fbcf3db61421b9990b9 Hello world ! 97 : 000ba61ca46d1d317684925a0ef070e30193ff5fa6124aff76f513d96f49349d
And here we are. Now we just have to let the network know our block, made of "Hello world" and 97, won.
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.