Gravitational force is known to exist between all bodies in the univer...
The gravitational interaction of the Earth and Moon produces a number of effects. The most obvious of these is the orbital motion of the Moon around the Earth, but there is also a motion of the Earth around the center of mass of the Earth-Moon system (discussed below), tidal effects on the Earth and Moon (to be discussed in Gravitational Effects of the Earth and Moon: Tides), and precessional effects on the Earth's axis of rotation (to be discussed in Gravitational Interactions of the Earth and Moon: Precession.
The Gravitational Force of the Earth and Moon on Each Other
According to Newton's Third Law of Motion, the Law of Action and Reaction, if the Earth exerts a force on the Moon, the Moon must exert an equal and opposite force on the Earth. Newton's Law of Gravity implies the same thing, as its mathematical formula,
F = G m M / r2
in which F is the gravitational force between the two bodies, m and M are their masses, and r is the distance between them, has the same value regardless of which object is thought of as pulling on the other.
In other words, both Laws imply that the force the Earth exerts on the Moon is numerically identical to the force the Moon exerts on the Earth. The only difference is that the Moon pulls the Earth toward the Moon, while the Earth pulls the Moon toward the Earth -- that is, the force on each object is toward the other object, and therefore in the opposite direction. (This is usually expressed by saying that the forces are "equal and opposite", but sometimes people think this means they cancel each other, unless it is made clear that the forces are acting on different objects.)
The Basic Effect of the Force Between the Earth and Moon
According to Newton's Second Law of Motion, the Force Law, the effect of a force on an object is to accelerate it in the direction of the force, according to the formula
F = m a
where F is the force applied to the mass m, and a is the acceleration.