The full name of the FA Cup is the Football Association
Challenge Cup and is the world's oldest association football competition.

The first FA Cup was held in 1871. At the time there were 50 clubs in the
association though only 15 of them applied to play in the first tournament and six of those changed their minds, leaving only nine clubs altogether. The FA
itself is even older - it goes back to 1863, though for the first years it
didn't hold any official events but just organised friendly matches between
members.

The first FA Cup final was between Wanderers and Royal Engineers, with
Wanderers eventually winning 1-0.
Not many of the clubs that played in the first FA Cup are still around or still have the same names, but a few are the same including Queen's Park Rangers and the Royal Englineers. Today over 700 hundred clubs are in the FA.
You can learn more about the FA Cup's history here.