A contingency is a condition in your purchase agreement that gives you the right to back out (and get your earnest money back) if certain things don't happen. The three most common are the financing contingency, the inspection contingency, and the appraisal contingency.
In a competitive market, buyers sometimes waive contingencies to make their offer more attractive. This is risky — if your loan falls through, the inspection finds problems, or the appraisal comes in low, you could lose your earnest money or be forced to close at terms you can't afford.
Talk to your agent and loan officer about which contingencies make sense to keep and which (if any) are safe to waive based on your specific deal.