Usually contained in an invitation to an event, an RSVP can be properly responded to by simply informing the requester of one's intentions to attend the event or not. RSVP is an abbreviation for the French phrase "répondez, s'il vous plaît," which translates to "please reply" in English. RSVP is often followed by another abbreviation--ASAP--meaning "as soon as possible," to encourage the invited party to respond as soon as possible.
How one responds can be as important as the reply itself. If the invitation is a wedding invitation and is sent via mail, it is courteous to also mail the reply; sometimes a response card is even included with the invitation. If the invitation is extended through email or a text message, however, it is usually perfectly polite to respond likewise.
It is important to note that if one intends to bring a guest to an event to which one has been invited to, that person must take care to give the host the guest's name and ask permission first.
Finally, it is common courtesy to reply in a timely manner to an RSVP, well before the actual event. Of course, situations can arise where one is not able to follow through on an accepted invitation, and it is vital to give the host as much notice as possible of any unplanned absence.