This law concerns two parts...
An object accelerates in the direction you push it and the harder you push it the more it will accelerate. That's pretty basic and seems like common sense. This law applies to both pushing and pulling so when I teach the Laws of Motion to 4th graders one "experiment" we do to show this is a tug of war. We create two sides and try to make it as balanced as possible to show that equal force creates balance. We tie a ribbon in the middle of the rope and have a line on the ground as well to show how the ribbon pretty much stays over the line.
We then create "unbalanced" teams so there is more force on one side. The rope is then pulled toward the side with greater force. It's pretty basic but it's a great physical and visual way to represent force and acceleration.
We'll recall the mathematical identity that describes the Newton's second law.
F = m*a
You can prove this law, conducting the following experiment: take two balls and put them on a plane table. This balls have equal masses. They are initially at rest. Push one ball harder than the other ball. You'll notice that the ball that has been pushed harder it's travelling faster than the other ball. Therefore, the higher the pushing force is, the more accelerated the moving of the ball is. Since their masses are equal, the force and the acceleration involved are directly proportional.