In short, a Browser Game is a game you play using a web browser. These can be played anywhere you have an internet connection, as virtually all internet capable computers have web browsers pre-installed. The web browser is a program which is used to download and view web pages.
The benefit to you is that you (normally) don't have to buy a new game system or install software to play. The benefit to us is that we don't have to mess around with shrink wrap and store shelves, and we can add new features without everyone having to purchase a second copy.