When a great force is applied to an object, what happens?

When a large force is applied to an object, it will accelerate according to Newton's second law of motion:

F = m x a

Thus, an applied force of F newtons will cause an object of mass m kg to move with an acceleration of a m/s^2. 

If the object was already moving, the direction of force will dictate whether the object accelerates or decelerates (negative acceleration). For example, if a car is moving and we apply large force in the opposite direction, the car will stop or slow. However, if the force was applied in the same direction as the motion, the car will accelerate.

Also, there may be cases when the force will not result in any motion at all. Imagine pushing against a wall or pushing a book down on a table. In these cases, neither the wall nor the book will move, but depending on the magnitude of force, you may end up breaking the wall or the table.

Hope this helps.