Acceleration is produced when a force acts on a mass. The greater the mass (of the object being accelerated) the greater the amount of force needed (to accelerate the object).
In the pictures below, when the soccer ball is kicked, there is enough force to move it, and therefore it accelerates. However, when the brick wall is kicked, there is not enough force in the kick to move it. Heavier objects require more force to move them the same distance as lighter objects, because the heavier object's mass is greater.