Bitcoin solves the "double spending problem" of electronic currencies (in which digital assets can easily be copied and re-used) through an ingenious combination of cryptography and economic incentives. In electronic fiat currencies, this function is fulfilled by banks, which gives them control over the traditional system. With bitcoin, the integrity of the transactions is maintained by a distributed and open network, owned by no-one.
What bitcoin miners actually do could be better described as competitive bookkeeping. Miners build and maintain a gigantic public ledger containing a record of every bitcoin transaction in history. Every time somebody wants to send bitcoins to somebody else, the transfer has to be validated by miners: They check the ledger to make sure the sender isn’t transferring money she doesn’t have. If the transfer checks out, miners add it to the ledger. Finally, to protect that ledger from getting hacked, miners seal it behind layers and layers of computational work—too much for a would-be fraudster to possibly complete.
Some nodes are mining nodes (usually referred to as "miners"). These group outstanding transactions into blocks and add them to the blockchain. How do they do this? By solving a complex mathematical puzzle that is part of the bitcoin program, and including the answer in the block. The puzzle that needs solving is to find a number that, when combined with the data in the block and passed through a hash function, produces a result that is within a certain range. This is much harder than it sounds.