The term R.S.V.P. stands for a French phrase, which means "Respond, if you please" (respondez s’il vous plait). Used mostly on written invitations, this cues the recipient of the invitation that the sender wants to know if the guest is planning to come or not. Based on how many people send back a reply that, yes, they're coming, the one issuing the invitation will know how many portions of food to order for whatever event he or she is planning. In addition, at a formal wedding, the host is then able to arrange the seating at the tables according to the people who informed him that they're going to attend.
In most Western cultures, it’s proper to try to respond as soon as you can. Wedding invitations usually come with a response card, wherein you can write your name on a line and indicate whether you will/will not attend on the next line. Sometimes a telephone number is included on an invitation for you to R.S.V.P. by phone. If you get an invitation with "regrets only" at the bottom, it means that your host is going to assume that you are coming, unless you call to tell him that, regretfully, you aren't able to.
The reason we use the initials of a French phrase on an English invitation is because many of the rules of etiquette that the Western world follows come from the court of King Louis XIV of France. These rules were either posted on signs in the palace or written on the back of invitations to court events. In the 19th century, French was considered a refined, high-class language in the United States, and many of the customs of France were adopted by those in America who aspired to high society.