Cryptocurrency mining is the process of using computers to solve "blocks" on a blockchain, with cryptocurrency being produced as a reward.
Mining is generally processor intensive, since a good deal of mathematical work is required to solve each block. This is typically a deliberate design feature of a cryptocurrency system, with "proof of work" being required to generate rewards. This "proof of work" helps give the currency value by making it difficult to produce (in much the same way as mining gold or any other commodity is valued to some extent by how difficult it is to get out of the ground). As is the case with mining gold or any other commodity there are overheads and capital costs associated with cryptocurrency mining. Typical overheads include the cost of electricity and systems maintenance, while capital costs include equipment and premises.
The first cryptocurrency to be mined was Bitcoin, and in the early days general purpose desktop computers could be used for the process. Competition quickly made this approach unviable, as it became clear that purpose built mining "rigs" would be much more suited to the process. [N.B. A mining rig is a computer which has been designed and built specifically for the purpose of mining cryptocurrency, typically using rows of individual and separately running processor units (called "cards") mounted in a frame and controlled by a single central processing unit on a motherboard.]
Even with a purpose built mining rig individual blocks are very difficult to solve, and commercial mining operations usually involve multiple rigs all working together in collective "farms". Each machine is connected to the Internet through which these farms and individual miners can collaborate in mining "pools". The idea behind a mining pool is that participating miners work together to solve each block, with the collective rewards being shared out whenever a block is solved by any miner in the group. This approach secures a stream of frequent regular payments rather than an infrequent (and unpredictable) "windfall" for each miner who would otherwise receive payment only as and when a particular block was (exclusively) "solved" by an individual machine. This "pooling" of mining resources effectively spreads payments out more evenly over time, thus helping with easing the pressure on each farm and individual miner's cashflow as required for covering ongoing costs such as e.g. electricity.
Besides Bitcoin many (but not all) cryptocurrencies can be generated by mining, and a whole new industry has sprung up around building, hosting and servicing mining rigs besides the farms doing the mining themselves. One popular choice is ether [ETH], the "official" currency of the Ethereum project, which can be mined using machines built around graphics processing cards as commonly used in computer gaming machines. Ether generated by mining can used as "gas" to power operations on the Ethereum Virtual Machine (such as e.g. running "smart contracts" etc.) or to purchase goods and services elsewhere (ether being widely used like Bitcoin as a cryptocurrency in its own right). It is planned, however, that Ethereum will move from "Proof of Work" to a "Proof of Stake" model, from which point on new "ether" [ETH] will no longer be generated by mining in the "traditional" sense.
Other cryptocurrencies which can be mined include "classic" ether [ETC] (a "spin off" coin which was created by a "hard fork" split in the Ethereum blockchain), Monero, Musicoin and Zcash - though there are many others in what is turning out to be a fast moving and rapidly evolving field!
[N.B. For more information please see the separate page about the blockchain, cryptocurrencies and the cryptoeconomy.]