Often written R.s.v.p. This is an initialism of Répondez s'il vous plaît, which is French for 'Respond if you please'.
This is often found on invitations to fancy-type events. Despite its literal translation, etiquette requires that you do respond, either to confirm that you will be attending or give your regrets that you cannot.
Unless otherwise specified, you should respond in writing. Even a very short note is fine. If 'RSVP' is followed by a telephone number or e-mail address, this is a sign that that is the proper way to respond. You should never RSVP in person.
You may also see 'RSVP by (Date)', or 'RSVP NLT (Date)'. This gives a formal deadline by which you should reply. It would be impolite not to respond by this date, and more so to respond in the affirmative after this date. (Worst would be to fail to respond and then show up anyway. Don't do that). If no date is given, you should respond within 24-48 hours of receiving the invitation.
You may also see 'RSVP regrets only'. In this case you should only respond if you are not going to be able to attend.
RSVP may be used as both a verb ("Please RSVP") and a noun ("I've received your RSVP"). It's not uncommon to see the forms 'RSVPed' (past tense), 'RSVPing', and 'RPVP's' (plural). Despite this being a French phrase, it is pronounced by spelling out the English letters, 'Ar-Ess-Vee-Pee'.