If you do something fitfully, you do it in restless or irregular way. When you're nervous about a big test the next day, you might sleep fitfully, tossing and turning.
If you've ever encouraged a group of preschoolers to sit quietly in a circle, you've noticed that most of them do it fitfully, switching back and forth between being completely distracted and paying attention. The adverb fitfully, which is great for describing this kind of on-and-off behavior, movement, or feeling, comes from the adjective fitful, which was used originally by Shakespeare to mean "suffering fits" and then by Romantic poets in the 18th century to mean "changing and shifting."