R.S.V.P. stands for a French phrase, "répondez, s'il vous plaît," which means "please reply." The person sending the invitation would like you to tell him or her whether you accept or decline the invitation. That is, will you be coming to the event or not? Etiquette rules followed in most Western cultures require that if you receive a formal, written invitation, you should reply promptly, perhaps that same day. For hosts who are planning a dinner party, a wedding or a reception, this is important from a practical point of view, because they need to know how many people to count on and how much food and drink to buy. More important, though, is the simple courtesy of responding to someone who was nice enough to invite you, even if it is to say that you regret that you will not be able to attend.
Many wedding invitations come with a response card that you can mail back right away. Other written invitations will carry the host's telephone number so you can call with your reply, although under strict etiquette rules, a written invitation requires a written reply. Nowadays, invitations often carry a "regrets only" notation at the end. That means that the host will count on your being there unless you tell him or her otherwise. Some people even use "R.S.V.P." as a verb, as in "Have you R.S.V.P.ed to that invitation?"
You might wonder why we use the initials of a French phrase in an invitation that is written in English. The answer you seek is on the next page.