Battledore and shuttlecock is a game which probably developed in Ancient Greece around 2000 years ago.
From there it spread East to China, Japan, India and Siam. Peasants played it in medieval England and by the late 16th century, it had become a popular children's game. In the 17th century, Battledore or Jeu de Volant was an upper class pastime in many European countries. Battledore and Shuttlecock was simply two people hitting a shuttlecock backwards and forwards with a simple solid bat as many times as they could without allowing it to hit the ground.
In 1830, the record for the number of hits was made by the Somerset family and was apparently 2117 hits. The origin of the shuttlecock is unclear although one theory has it that corks were used to store feathers for some reason (writing feathers perhaps) in the past and the resulting object became a popular thing for first throwing and then batting.
Badminton is a direct descendant of Battledore and Shuttlecock The games origin is therefore traceable to the East, but the name 'badminton' comes from Badminton House, the Duke of Beaufort's residence in Gloucestershire (now Avon) where a new version of battledore had emerged by the end of the 1850's. (Isaac Spratt, a London toy dealer published a booklet, "Badminton Battledore - a new game," in 1860, but unfortunately no copy has survived.).
When Battledore became Badminton isn't known exactly but the first rules of the game were written in Poona, India by the British in 1873 and these were not dissimilar to the modern game. The game was introduced into New York by two gentlemen, in 1878. One returning from India and the other from England. The game (which was sometimes known as 'hit and scream') was very popular there as well as in England during the 1870's and 1880's but more as a social outdoor pastime rather than a sport.
Sources: 'Better Badminton' (Jackson & Swan 1939), www.tradgames.org.uk