Mining is the process of adding transaction records to Bitcoin’s public ledger of past transactions (and a “mining rig” is a colloquial metaphor for a single computer system that performs the necessary computations for “mining”. This ledger of past transactions is called the block chain as it is a chain of blocks. The blockchain serves to confirm transactions to the rest of the network as having taken place. Bitcoin nodes use the blockchain to distinguish legitimate Bitcoin transactions from attempts to re-spend 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 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.