When you do something fairly, you treat everyone with equal consideration. Sharing a pizza fairly between six people is easy: just give each person one slice.
A teacher who doesn't treat all students fairly isn't a very good teacher — and a police officer who doesn't act fairly toward all citizens is also doing a poor job. When you behave fairly, you're unbiased and impartial, and you follow the rules. You can also use this adverb to mean "to a large degree" or "reasonably." For example, you could say, "I'm fairly certain that it's going to snow again tomorrow."