A bitcoin faucet is a reward system, in the form of a website or app, that dispenses rewards in the form of a satoshi, which is a hundredth of a millionth BTC, for visitors to claim in exchange for completing a captcha or task as described by the website. Faucets usually give fractions of a bitcoin, but the amount will typically fluctuate according to the value of bitcoin.
Some faucets also have random larger rewards. To reduce mining fees, faucets normally save up these small individual payments in their own ledgers, which then add up to make a larger payment that is sent to a user's bitcoin address.
Faucets can help introduce new people to bitcoin or to altcoins. A majority of faucets provide information to new users as well as offering them some free coins so that they can "try before they buy", experimenting with a test transaction or two before putting real money on the line. Since this whole experience is so new and a bit complicated to people, this is a beneficial way to promote digital currency and bring in new users.