There are several forces involved. As stated, gravity is a major one. All objects that have mass have a gravity directly associated with that mass. the larger the mass, the more the gravity. The force that pulls two objects in the same direction is relative to the amount of mass each object has.
The next major "force" is inertia. That is the tendency of an object in motion to stay in motion. You can see these two forces when you through a ball. The inertia is what you impart on the ball and the gravity is what pulls it to the ground.
There is also electromagnetic forces that impart a force on planets (sometimes referred to as solar winds). You can actually visualize this as the northern lights or the aurora borealis as this energy is absorbed by the Earth's magnetic field and excited electrons that then emit photons.