A head-strict function will not necessarily evaluate every cons cell of its (list) argument, but whenever it does evaluate a cons cell it will also evaluate the element in the head of that cell. An example of a head-strict function is
beforeZero :: [Int] -> [Int] beforeZero  =  beforeZero (0:xs) =  beforeZero (x:xs) = x : beforeZero xs
which returns a list up to the first zero.
This pattern of evaluation is important because it is common in functions which operate on a list of inputs.
See also tail-strict, hyperstrict.