A border is an outer edge around something like a puzzle or a country. Add an "a" and a boarder is a person who pays to live and eat somewhere, like boarding school.
The word border comes from an Old French word meaning "a colorful band that goes around a shield." You don’t need a shield to use the word: draw a line around a picture and you just made a border. Or perhaps you’d like to border your collar with rhinestones. The border around a country is invisible, but hard to cross without a passport. Here are some examples:
Decorative hieroglyphic borders were added to the poster-paint pictures that already adorned the walls of the temple. (The Egypt Game)
The whole situation was a little suspicious, but the girl's passport was in good order, and she'd been approved to cross the border before. (A Wish in the Dark)
The word boarder is related to board, as in a plank of wood. The meaning evolved into something like "a table," then boarder took on the meaning of someone who eats at a table. And now the word refers to a person who pays to eat and stay somewhere. If you have a big house, you might take on a boarder to help pay the bills. Here are some examples:
A man named Klipspringer was there so often and so long that he became known as "the boarder"—I doubt if he had any other home. (The Great Gatsby)
Some children are boarders at the school but, for now, the Cambridges will be day pupils. (BBC)
To remember the difference, note that the word boarder has that extra "a'' renting a room in the word.