When you err, you make a mistake or do something wrong. When a newspaper reporter errs in a printed story, the paper often prints a correction the next day.
By itself, err, which shares a Latin root with error, is a formal way to say "mess up." The proverb "To err is human, to forgive divine" is an old-fashioned version of "Hey, everyone makes mistakes. Why don't you be the bigger person and let it go?" Err can also mean to go in a certain direction, as in another common saying, "Err on the side of caution," or "Play it safe."