Lekku were the head-tails of a Twi'lek or Togruta, which contained a part of the brain and excess fat in the former species, and useless decorations in the latter. Lekku were the Twi'leks' most notable feature, and played a major role in their culture.
Twi'leks could communicate with their lekku using a silent language of small twitches and movements. Lekku didn't have a lot of muscle or fine control, so Twi'leks couldn't say, grab a doorknob with them or use chopsticks with them, but they could twitch and nudge and tickle with them, which led to the other main use of lekku in Twi'lek culture: sex play. Lekku fit nicely into any orifice, they could wrap around body parts, rub against them... hey, the possibilities are endless, as evidenced by the special niche market of Twi'lek lekku sex guides on amazon.com, including Lek Me, Lek Me, Lek Me.
Some Twi'leks became so well-nourished (fat) that they grew a second set of lekku, which was an aristocratic status symbol despite looking very disgusting. See Orn Free Taa. Ahsoka Tano, unlike the rest of her species, had no lekku; this could be due to any number of reasons, including birth defect or unfortunate lightsaber incident.