In framing carpentry for wood-framed buildings in the U.S., a cripple stud is a wall stud that is shorter than the full length, which is from floor to ceiling. Cripple studs are used when a large hole is made in a wall for a good and intentional purpose, mostly for a door or window.
Let's say you decide to put a door in a completed wall, so you take a big saw and cut a hole in the wall a few inches bigger than the door on all sides. Because the door is wider than the 16 inches or so between the studs that run from the floor to the ceiling, you end up with one or more short studs that hang down from the ceiling to the top of the hole. Those are cripple studs. Now, if you had cut out a window hole instead of a door hole, you'd have cripple studs both above and below the hole.
The purpose of the studs in bearing walls is to support what is above them, which may be ceiling joists or the joists of an upper floor., so cutting studs off short weakens the structure and puts you in violation of the building code. The solution is to put something under the cripple studs that are now dangling from the ceiling to support them. That something is a horizontal beam of adequate strength to carry the weight that was originally carried by the studs before they were crippled. Cool people know that this horizontal support is called a lintel or header. The lintel is supported on both sides by another type of short stud called a jack stud or trimmer. The jack studs are sistered with the full-length king studs on both sides of the opening. For a window, the framing is completed by a horizontal beam placed across the lower cripple studs. That beam is called a sill. In window framing, the jack studs rest on the sill and support the header beam.
Of course there's more to making a door or window than just cutting holes in walls, but now, at least, you know what a cripple stud is.