Nearly everything in blender can be equiped with a keyframe. This includes properties like alphachanel, etc. You can use it for things getting invisble, etc. Properties can be marked with a keyframe via pressing with the right mouse button onto it
First add a path or Nurbs object, then add as contraint to the object you want to move: follow path.
If you run the animation you will see that nothing happens. This is because of the missing keyframes.
Select the path object, go to the property path animation. There is property named “Evaluation time” with value 0. Rightclick it and add a keyframe. Go forward in timeline then set the evalution time on the value of the property frames and add another keyframe. You will see that the object moved and you will see it moving if you press play