What the appraisal does
A lender lends against the lower of the contract price or the appraised value. The appraisal protects both sides from a loan that exceeds what the market supports. It is ordered by the lender and performed by an independent state-licensed appraiser, and the borrower normally pays for it.
When it comes in low
Suppose you offer $500,000 and the appraisal returns $485,000. The loan is sized from $485,000, leaving a $15,000 gap. With an appraisal contingency you have options: renegotiate the price, split the difference, cover the gap in cash, or walk away with your earnest money.
Without the contingency you must cover the gap or risk your deposit. That is the trade you accept when you waive it to compete.
Challenging a low appraisal
A reconsideration of value can work when the appraiser genuinely missed a comparable recent sale. Your agent supplies the comps and the lender submits the request. It is not a negotiation over the number, it is an evidence submission, and it often does not change the result.
A practical approach
Keeping the contingency is the lower-risk choice in all but the most extreme bidding situations. If you feel you must give something up, capped gap coverage, meaning a clause agreeing to cover an appraisal shortfall up to a stated dollar amount, competes nearly as well as a full waiver while limiting how much you are exposed to.
Before you agree to any gap figure, check what it does to your cash position using /mortgage-calculator/purchase/.