Whilst is defined as while and refers to things going on at the same time as something else.(conjunction)
If you fall asleep as you are watching a movie, this is an example of a situation where you fall asleep whilst watching a movie.
See whilst in Webster's New World College Dictionary
Origin: ME whilest, extended < whiles, whiles
See whilst in American Heritage Dictionary 4
Origin: Middle English whilest
Origin: , alteration of whiles, whiles; see whiles.