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 EarthMoon 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 / r^{2}
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, or rate of change of the velocity of the object. As written here the formula does not directly show it, but the force and acceleration are vectors, meaning that they have both magnitude and direction. In this case, since they are the only vectors in the equation, they must have the same direction  that is, the change of velocity must be in the direction of the force, as stated above.
If the Earth and Moon did not exert a force on each other they could each move indepently of the other, but because they do exert a force on each other, their velocities are changed according to the magnitude and direction of each force and their respective masses. Since each is pulled toward the other, the Earth is pulled toward the Moon and therefore a little away from the path it would otherwise follow around the Sun in the absence of the Moon, and the Moon is pulled toward the Earth and therefore a little away from the independent path it would otherwise follow around the Sun in the absence of the Earth. If one or the other did not exist, the remaining object would orbit the Sun in an orbit nearly identical to the path the pair currently follows around the Sun, but since both exist they each follow a path that is roughly the same as their imaginary independent paths, but not quite the same paths as a result of their interaction with each other.
The forces that the Earth and Moon exert on each other must be equal, but that does not mean that the effects of those equal forces are the same, because the two objects have very different masses. The Earth is 81.6 times more massive than the Moon, and as a result a given force will affect (or change) its motion 81.6 times less than it would the Moon. The most obvious effect of the Earth's pull on the Moon is that the Moon orbits the Earth once every 27.3 days, moving in an elliptical path with a size of about 240,000 miles. The far less obvious effect of the Moon's pull on the Earth is that the Earth also "orbits" the Moon every 27.3 days, with an elliptical path 81.6 times smaller than that of the Moon, or only 3,000 miles in size.
Of course the Earth can't possibly be orbiting the Moon at a distance of 3,000 miles while the Moon orbits the Earth at a distance of 240,000 miles. So the elliptical paths just described cannot be around each other. Instead, what the two bodies actually orbit is a point called the center of mass (or barycenter) of the EarthMoon system, with an orbital path around that point of 3,000 miles size for the Earth, and 240,000 miles size for the Moon. The principle is exactly like that of a seesaw or teetertotter. If two people of very different weights sit on opposite sides of the balance point (or fulcrum), the heavier person must sit closer to the balance point in inverse proportion to their relative weights. For instance, if the heavier person weighs twice as much as the lighter one, they must sit twice as close to the fulcrum. The balance point is the "center of mass" of the seesaw, just as the barycenter is the balance point of the EarthMoon system. It is this point that actually moves around the Sun in what we call the orbit of the Earth, while the Earth and Moon each move around the barycenter in their respective "orbits".
Some Notes About the Barycenter
In measuring the distance between two gravitationally interacting objects, we use the center of each body as the starting point for that measurement. Since the Earth's radius of 4,000 miles is larger than the 3,000 mile distance between the Earth and the barycenter of the EarthMoon system, the barycenter is actually inside the Earth, a thousand miles beneath the place where the Moon is overhead. Not that it makes any physical difference exactly where it is, because it is a strictly imaginary point used only to divide the Earth's complicated motion (mostly due to the Sun's gravity, but partially due to the Moon's) into two less complicated pieces  the elliptical motion of the barycenter around the Sun, and the (much smaller) elliptical motion of the Earth around the barycenter.
In the discussion above it was mentioned that the Earth is 81.6 times more massive than the Moon. Nowadays we can determine this precisely by measuring the gravitational effect of each object on manmade satellites orbiting either body. But less than a hundred years ago this would have been impossible; and yet by the 1700's it was possible to tell that the difference in their mass was in this range. As the Earth moves around the barycenter every 27.3 days, telescopes pointed at other planets will see them moving around a 6,000 mile wide path centered on their predicted paths once every 27.3 days. This deviation in their positions is relatively easy to measure, and realizing that it is due to our motion around the barycenter of the EarthMoon system, a comparison of the size of the deviation to the size of the Moon's orbit made it possible to determine the relative masses of the Earth and Moon.
Finally, since both the Earth and Moon move around the barycenter, the average distance between them, which is called the semimajor axis of the Moon's orbit, is not the same as the Moon's real motion, but is actually the combined motions of both the Earth and the Moon around the barycenter. It is for this reason that Newton's version of Kepler's Third Law of Planetary Motion introduces the combined mass of the two bodies as a part of its mathematical formula:
P^{2} = 4 π ^{2} a^{3} / G (M + m) 
(This page is in the process of being converted from rough lecture notes to a selfcontained but hopefully clearer discussion. Part of what follows, which is mostly a less mathematical discussion of what is covered above, will be retained either as the separate discussion below or incorporated into the discussion above, while other portions will be removed as needlessly repetitive.)
Another Look at the Earth's Barycentric Motion
Just as the Moon moves around the Earth once every 27 1/3 days as a result of the Earth's pull on the Moon, the Earth moves "around the Moon" once every 27 1/3 days as a result of the Moon's pull on the Earth. More accurately, each moves around a point in between them, which would be the balance point between them if they were on a seesaw, called the center of mass or barycenter of the EarthMoon system.
At any given time, the bodies are on opposite sides of the center of mass, moving in opposite directions. As shown in the (currently nonexistent) diagram below, each exerts a force on the other which, according to Newton's Third Law of Motion (the Law of Action and Reaction), is equal and opposite to the force that the other is exerting on it; but although the forces are equal, their effects are not, because the more massive Earth is accelerated less by the force than the less massive Moon is.
*diagram to be inserted here*
The Earth and Moon are each moving around the center of mass of the EarthMoon system. Since the Earth is much more massive, it is much closer to the center of mass, which is actually inside the Earth.
According to Newton's Second Law of Motion (the Force Law), the same force acting on different masses will produce accelerations which are inversely proportional to the mass; so the 80 times more massive Earth is accelerated 80 times less than the Moon, and must move around a path which is 80 times smaller than the Moon, with the center of mass at the focus of their respective motions. Since the semimajor axis of the Moon's orbit is about 240,000 miles, the Earth's motion around the center of mass must be about 3000 miles in size. This is less than the 4000 mile radius of the Earth, so the center of mass is actually inside the Earth, 1000 miles below the place where the Moon is directly overhead.
As the Earth rotates to the east each day, the Moon appears to move to the west, and the place where the Moon is overhead moves west as well. This has no physical effect, because the center of mass is not a real physical object, but an imaginary point defined so the complex motion of the Earth and Moon are easier to visualize and understand. Each body moves around the Sun in a path which is almost identical to the elliptical orbit of the Earth, but weaves back and forth relative to that path every 27 1/3 days due to their mutual interaction. This complex motion can be simplified by breaking their motion into two parts: the 27 1/3 day motion around the center of mass of the EarthMoon system, and an annual motion of that point around the Sun.
Note that whether the Moon exerted any force on you or not, you would still be compelled to move around the center of mass every month, because the Earth's gravity holds you on the surface while it goes around the center of mass; but you aren't just along for the ride, as the Moon is pulling on you with a force equal to 1/300,000th of your weight, in a way that would cause you to follow the path that the Earth makes around the barycenter even if you weren't firmly held to the Earth by its gravity. The fact that the Moon is actually pulling on you separately from its pull on the Earth means that depending upon where you are on the Earth, the force acting on you may be greater than or less than the average force of the Moon on various parts of the Earth. It is that differential force that leads into the next page in this discussion, Gravitational Interactions of the Earth and Moon: Tides
