Mining is intentionally designed to be resource-intensive and difficult so that the number of blocks found each day by miners remains steady. Individual blocks must contain a proof of work to be considered valid. This proof of work is verified by other Bitcoin nodes each time they receive a block. Bitcoin uses the hashcash proof-of-work function.
The primary purpose of mining is to set the history of transactions in a way that is computationally impractical to modify by any one entity. By downloading and verifying the blockchain, bitcoin nodes are able to reach consensus about the ordering of events in bitcoin.
Mining is also the mechanism used to introduce Bitcoins into the system: Miners are paid any transaction fees as well as a "subsidy" of newly created coins. This both serves the purpose of disseminating new coins in a decentralized manner as well as motivating people to provide security for the system.
Bitcoin mining is so called because it resembles the mining of other commodities: it requires exertion and it slowly makes new units available to anybody who wishes to take part. An important difference is that the supply does not depend on the amount of mining. In general changing total miner hashpower does not change how many bitcoins are created over the long term.
The Computationally-Difficult Problem
Mining a block is difficult because the SHA-256 hash of a block's header must be lower than or equal to the target in order for the block to be accepted by the network. This problem can be simplified for explanation purposes: The hash of a block must start with a certain number of zeros. The probability of calculating a hash that starts with many zeros is very low, therefore many attempts must be made. In order to generate a new hash each round, a nonce is incremented. See Proof of work for more information.
The Difficulty Metric
The difficulty is the measure of how difficult it is to find a new block compared to the easiest it can ever be. The rate is recalculated every 2,016 blocks to a value such that the previous 2,016 blocks would have been generated in exactly one fortnight (two weeks) had everyone been mining at this difficulty. This is expected yield, on average, one block every ten minutes.
As more miners join, the rate of block creation increases. As the rate of block generation increases, the difficulty rises to compensate, which has a balancing of effect due to reducing the rate of block-creation. Any blocks released by malicious miners that do not meet the required difficulty target will simply be rejected by the other participants in the network.
When a block is discovered, the discoverer may award themselves a certain number of bitcoins, which is agreed-upon by everyone in the network. Currently this bounty is 6.25 bitcoins; this value will halve every 210,000 blocks. See Controlled Currency Supply.
Additionally, the miner is awarded the fees paid by users sending transactions. The fee is an incentive for the miner to include the transaction in their block. In the future, as the number of new bitcoins miners are allowed to create in each block dwindles, the fees will make up a much more important percentage of mining income.