Each miner (or the pool if the miner is part of a pool) will set a policy on the conditions under which they will confirm a transaction. Generally speaking a miner will confirm a TX without a fee if it is under a certain size (in terms of blocksize), they may allow for a greater size if the inputs have not been spent in a long time and are sufficiently large (in terms of amount being transferred). Some miners will also give priority to a TX that has been unconfirmed for a long time.

The process is automatic and is setup via the software that "assembles" the block when it is sent to the rest of the network when it is found.