Gravity is a force. It is a function of the mass of an object(the greater the mass, the greater the gravitational force.) The strength of the force diminishes according to the distance from the object, so two identical objects would exert different gravitational force depending on their distance, the further away the weaker.
When an object "falls" under the influence of gravity, it accelerates towards the object. We usually are referring to a relatively small object and a relatively large one, IE an apple falling towards the ground. When it hits the ground, of course, it stops accelerating, but gravity is still exerting its influence on it. As it does on us when we are standing on the ground. If it were possible to remove the ground from under our feet, we would indeed continue our journey towards the centre of the earth.
The gravitational pull at the poles is in fact slightly stronger than it is at the equator. This is due to the equatorial bulge, and the flattening at the poles, which means you are closer to the centre of the earth when you stand at the poles than you are at the equator, and the gravitational force is therefore stronger. This is why you generally get better performances and more new records at the Olympic Games when they are held closer to the equator.
The gravitational force is NOT caused by the spin of the earth. The movements of the planets are due to the effects of the original momentum of the matter from which they were made, though this has been affected over time, often due to gravitational effects. For instance, the rotation of the earth has slowed down over time due to the gravitational effect of the moon.