A boarder is someone who rents a room in someone's house. It can also be a student who lives and studies away from home at a boarding school.
There are two main meanings of boarder, but they both involve staying somewhere away from home. A boarding school is a private school where students live as well as study. Those students are called boarders. Also, if someone rents a room of their house to guests, the guests are boarders. Holden Caulfield, the main character of "Catcher in the Rye," was a boarder; that is, until he got thrown out of Pencey Prep.