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would have (auxiliary used to form the past conditional)
S'il avait su, il aurait refusé.
If he had known, he would have refused.
is said to have; reportedly did (used to report rumours or unconfirmed information)
Il aurait volé l'argent, d'après les voisins.
He is said to have stolen the money, according to the neighbors.
Notes
GENERAL INFO
aurait is the third-person singular conditional of the verb *avoir* (to have). It means "would have" (for *il/elle/on*) or "there would be" (for *il y aurait*).
USAGE
It expresses hypothetical situations or politeness.
Examples: "Il aurait faim" (He would be hungry). "On aurait pu y aller" (We could have gone there). "Il y aurait un problème" (There might be a problem / There would be a problem).
TRAPS
Do not confuse aurait with *aurez* (future tense of *avoir*, "you will have"). Also, remember it's part of the conditional perfect (e.g., *il aurait mangé* - he would have eaten).