An appraisal is a licensed professional's estimate of what a home is worth, ordered by your lender before it will fund the loan. The appraiser compares the property to recent nearby sales, adjusts for condition, size, and features, and delivers a formal opinion of value.
Lenders care because the house is their collateral: they lend against the appraised value or the purchase price, whichever is lower. If the appraisal comes in below your offer, the gap is your problem — you renegotiate the price, bring extra cash, challenge the appraisal with better comparable sales, or walk away (an appraisal contingency protects your deposit here).
The number also feeds your loan-to-value ratio, which affects your rate and whether you pay mortgage insurance. Don't confuse it with a home inspection: the appraisal protects the lender's money, while the inspection protects you from buying a house with hidden problems. You generally pay for both, but you only see the appraiser's report — the value — not a repair list.