Err is just a formal way of saying you've made a mistake or done something wrong. A married man might err by going on a date with another woman, but he will surely feel the repercussions of his wrongdoing when he gets home!
By itself, err means to make a mistake. The old proverb "To err is human, to forgive divine," is an old-fashioned way of saying, "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," which just means, "Play it safe."
