Is this for the Simply Prizes comp? It's a badly worded question since there was an official 'Royal Mail' as far back as 1516, but it literally was just 'mail for royalty'.
The first really 'public' Royal Mail system began when letter boxes were installed in 1840.
The Royal Mail traces its history back to 1516, when Henry VIII established a "Master of the Posts". The Royal Mail service was first made available to the public by Charles I on July 31, 1635, with postage being paid by the recipient, and the General Post Office (GPO) was officially established by Charles II in 1660.