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.
Backtracking a bit, let's talk about "nodes." A node is a powerful computer that runs the bitcoin software and helps to keep bitcoin running by participating in the relay of information. Anyone can run a node, you just download the bitcoin software (free) and leave a certain port open (the drawback is that it consumes energy and storage space – the network at time of writing takes up about 145GB). Nodes spread bitcoin transactions around the network. One node will send information to a few nodes that it knows, who will relay the information to nodes that they know, etc. That way it ends up getting around the whole network pretty quickly.
As soon as a miner finds a solution and a majority of other miners confirm it, this winning block is accepted by the network as the “official” block for those particular transactions. The official block is then added to previous blocks, creating an ever-lengthening chain of blocks, called the “blockchain,” that serves as a master ledger for all bitcoin transactions. (Most cryptocurrencies have their own blockchain.) And, importantly, the winning miner is rewarded with brand-new bitcoins (when Carlson got started, in mid-2012, the reward was 50 bitcoins) and all the processing fees. The network then moves on to the next batch of payments and the process repeats—and, in theory, will keep repeating, once every 10 minutes or so, until miners mine all 21 million of the bitcoins programmed into the system.