Back and forth: moving first in one direction and then in the opposite one. The kids played with the ball in the back yard kicking it back and forth between them. Backward can have various meanings: 1.Towards the back: Start the car and drive backward 2.Having learning difficulties (and if used it's meaning is offensive) 3.It is often used to describe a person/era/place that is not progressive (E.x. backward people) or up-to-date, and many more things. As for forward, it is the opposite of backward.

"Back and forth" is more general. It applies to anything going from one side to another and then returning, etc. it can be used literally or figuratively. The other relates more specifically to the directions "in front" and "behind". It is usually used literally.