The full form of RSVP is répondez s'il vous plaît
The phrase is French for "Reply if you please" or simply "Please reply", and is used at the end of formal letters which require a response from the reader, such as an invitation.
Interestingly, most French people would never use this phrase in a letter. Instead, they would opt for "Prière de Répondre".
RSVP came into fashion in England, in the Middle Ages, when use of the French language was considered "upper-class" etiquette (another French word).
This preference for French words among high society spread to the United States, and was fairly common until the late 19th century.