BAT, Basic Attention Token, is a cryptocurrency, like Bitcoin. A cryptocurrency is a token/point system which usually runs on top of a distributed database called a blockchain which, under ideal conditions, tracks the tokens that everyone using the currency has access to and makes it impossible (or, at least, impractically difficult) to access someone else's tokens, send someone else the same tokens twice (double-spend), or do other things you generally don't want people to do with money. People can use services called exchanges to trade their tokens for "real" currency like US dollars or vice versa.
BAT is integrated with the Brave browser. Web site operators, or webmasters, can mark their sites as accepting of BAT tokens. If they do so, Brave will track how long a user spends on that site and send a corresponding amount of BAT to the webmaster as "payment" for their usage of the site; you can also manually tip a desired amount to a webmaster directly. The BAT you send will either be purchased by you manually or allotted to you after you see ads sent to you by the Brave company.