Mining, or generating, is the process of adding transaction records to Bitcoin's public ledger of past transactions. This ledger of past transactions is called the block chain as it is a chain of blocks. The block chain serves to confirm transactions to the rest of the network as having taken place. Bitcoin nodes use the block chain to distinguish legitimate Bitcoin transactions from attempts to respend coins that have already been spent elsewhere.
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's public ledger (the 'block chain') was started on January 3rd, 2009 at 18:15 UTC presumably by Satoshi Nakamoto. The first block is known as the genesis block. The first transaction recorded in the first block was a single transaction paying the reward of 50 new bitcoins to its creator.
Bitcoin mining is so called because it resembles the mining of other commodities: it requires exertion and it slowly makes new currency available at a rate that resembles the rate at which commodities like gold are mined from the ground. See Controlled Currency Supply.
Mining contracts - provides mining services with performance specified by contract. An example would be where a specific level of mining capacity is rented out for a set price for a specific duration. Short list of mining contractors is Minerlease, 
Mining shares - provides Mining as a Service (MaaS) model; break large-scale datacenter mining down to easily manageable pieces that are available in the form of shares of equipment. Mining shares are available from such companies as ASICMINER or Bit-miner.com
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. It is recalculated every 2016 blocks to a value such that the previous 2016 blocks would have been generated in exactly two weeks had everyone been mining at this difficulty. This will yield, on average, one block every ten minutes. As more miners join, the rate of block creation will go up. As the rate of block generation goes up, the difficulty rises to compensate which will push the rate of block creation back down. Any blocks released by malicious miners that do not meet the required difficulty target will simply be rejected by everyone on the network and thus will be worthless.
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 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.
Users have used various types of hardware over time to mine blocks. Hardware specifications and performance statistics are detailed on the Mining Hardware Comparison page.
Early Bitcoin client versions allowed users to use their CPUs to mine. The advent of GPU mining made CPU mining financially unwise. The option still exists in the reference Bitcoin client, but it is disabled by default.
FPGA mining is a very efficient and fast way to mine, comparable to GPU mining and drastically outperforming CPU mining. FPGAs typically consume very small amounts of power with relatively high hash ratings, making them more viable and efficient than GPU mining. See Mining Hardware Comparison for FPGA hardware specifications and statistics.
An application-specific integrated circuit, or ASIC, is a microchip designed and manufactured for a very specific purpose. ASICs designed for Bitcoin mining were first released in 2013 and (at the time of this writing) are in the hands of a very limited number of miners. For the amount of power they consume, they are vastly faster than all previous technologies and already has made GPU mining financially unwise in some countries and setups.
As mining a block became more and more difficult, individuals found that they were working for months without finding a block and receiving any reward for their mining efforts. Thus they started organizing themselves into pools so that they could share rewards more evenly. See Pooled mining and Comparison of mining pools.
- Bitcoin Mining in Plain English by David Perry
- Tutorial to automatically start mining when you lock your computer. (Windows 7)
- Simplified Explanation of Bitcoin Mining by reddit user azotic