Sleep is an ailment that appears mostly in Mario RPG games. It is an aliment that Mario, his partners, and even enemies can get. This ailment doesn't appear in any of the Mario & Luigi games.
Sleep is commonly used by mushroom-related enemies. When Mario or his team gets this ailment, they rest for a number of turns unable to attack. Refreshing items and Peach's healing moves are the only ways to cure this.
Sleep is very common in the Paper Mario series most notably the first three games. Whenever Mario or his partners get this ailment, they are put to sleep for a few turns unable to attack. Crazee Dayzees are known for their only attack being to put Mario to sleep and is very tough to Guard against.
Mario himself can put enemies to sleep by using either a Sleepy Sheep, Sleepy Stomp, or using the Lullaby Star Power move. This ailment is greatly needed for the fight against Kent C. Koopa as he will put to Sleep for four turns and the success rate is very high.