What keeps the Moon up in the sky
The Earth attracts every body because of gravity. That's why when we throw a ball up, it falls down again. If gravity acts on the moon as well, why doesn't the moon fall into the Earth? The answer is also, gravity.
BOX on Gravity
Gravity is one of the earliest known forces. Any bodies that have mass interact with each other through gravity. From the days of Isaac Newton, scientists have been trying to find out more about gravity. In school, you might have read that the force between two masses, m1 and m2 is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
This means that the larger the mass, the greater the force. The further apart they are, the lesser the force, and the dependence on the "inverse square" distance means that if you move the masses twice (thrice) the distance apart, the gravitational force between them will become four (eight) times smaller (1/2^2=1/4 and 1/3^3=1/8). This means that the objects experience gravity even if they are placed very very far apart, with the gravitational force going to zero only when the objects are infinitely far apart.
However, the most important property of gravity is that it is always attractive. This is in contrast to electric or magnetic forces, for instance, which can be repulsive or attractive depending on whether the charges (or poles) are the same or different. Hence the gravitational force always brings bodies closer to one another, and that is why the ball falls back on Earth.
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Types of forces
What do forces do to an object? In short, you can say that forces CHANGE the motion of the object. 
Let us look at three different cases for forces.
1. Force pushing in the same direction as the velocity of the object. Suppose an object is moving to the left with a force pushing in the same direction. See the diagram. Let this force be constant. Then, since the force and object's velocity are in the same direction, the force makes the object speed up or accelerate. This is as per Newton's second law of motion, which states that objects change their momentum or accelerate when acted upon by an external force.
2. Force pushing in the opposite direction as the velocity of the object. This is almost the same case as above, but for an object moving to the right the force would be to the left.
Here the object slows down. Again, there is acceleration, but the slowing down is referred to as deceleration, like when you put on the brakes of a car.
3. Force pushing perpendicular to the velocity of the object. Let us call this a "sideways" force.
If it is just a sideways force, the object doesn't speed up and it doesn't slow down. It just turns. Of course, in order to exert a continuous sideways force, the force would have to point in a different direction as the object turns or it wouldn't still be "sideways". Here is an example. Take a ball at the end of a string - or maybe a yoyo since the string is already attached. Swing the ball around in a circle. Why does it move this way? The string pulls on the ball. But since the string can only pull in the direction of the string (you can't push with a string), the ball has a sideways force on it and changes direction. Such a sideways force that is pointing to the centre of the circle is called centripetal force.
Trajectories
What about the Moon, then? Why is it not subject to gravity? Or, is it? Let us first look at the action of a ball when it is thrown sideways. (Or you can think of a shotput or javelin throw).
The picture shows Barbora Spotakova of the Czech Republic on her way to winning the gold medal in the Women’s Javelin Throw Final on day seven of the 11th IAAF World Athletics Championships on August 31, 2007 at the Nagai Stadium in Osaka, Japan. Look at the 'angle of attack'. The distance the javelin travels depends on the angle at which it was launched or thrown, as can be seen from the figure. The important thing to note is that once the javelin has been released, the only force on it is the vertical downward force of gravity. The horizontal force which allows it to travel farther and farther (and hence win the gold medal for the most distant throw) is given by the athlete when it leaves her hands.
The shape of the path that the javelin travels is called its trajectory. The trajectory shown has a special name: it is called a parabola. Any object that is sent upwards with non-zero velocity and is acted upon only by gravity has a trajectory like a parabola, as can be seen from the picture of the water fountains. The larger the initial velocity, the farther the object travels.
The figure shows an example of a bullet being fired by a tall man (about 1.7 m above the ground). For a velocity of about 1.4 km/s, which is really fast for a bullet, it would travel almost 1 km (0.84 km) before it hits the ground. If the same rifle were fired from a balloon 34 km above the earth, it would travel 119 km before it hit the ground. As the velocity is increased more and more, the bullet would travel farther and farther, until it starts to "miss" the ground. At this critical velocity, it simply goes into orbit around the Earth.
This is shown in the picture of Newton's cannon. The moon does the same thing: it is falling towards the Earth so fast that it actually misses! In fact, all satellites do the same thing. The International Space Station (ISS) is orbiting the Earth at a mean velocity of about 7.6 km/s or 27,600 km/hr. If the velocity is increased even more then the object can "escape" from Earth's gravity and enter space. That is how space ships are launched. This escape velocity is about 11.2 km/s.
Circular versus Elliptical Trajectories
One last point. The moon's trajectory (and in fact the Earth's trajectory aound the Sun) is not exactly circular, but a squashed circle, called an ellipse. Such a trajectory arises when the force is not exactly pointing to the centre of the orbit but is off-centre. So the force has a component perpendicular to as well as in the direction of the velocity (see fig). In this case, the object would both speed up AND change directions. Because of this the moon does not have a constant velocity as it orbits the Earth.
The moon moves closer to the Earth and speeds up as it does so. As the moon moves away from the Earth, the opposite happens. This is part of the reason behind the Super Moon that occurred a little while ago.
BOX on Earth-Moon distance
Look at the actual sizes and distances of the Earth and Moon in the picture. It is not drawn this way in text books because it is difficult to see. So the picture is usually not drawn to scale, as in the second figure where the moon is only 1/5^th the distance it is suppose to be (but the correct relative size).
In the figure, the arrow represents the gravitational force on the moon. If the moon were moving in a perfect circle, the gravitational force would always be "sideways" and just cause it to change its direction.
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What about Moon's gravity?
Newton's third law states that action and reaction are equal and opposite and act on different bodies. This means that if Earth pulls the moon towards itself, the moon should do the same thing: it should pull the Earth towards itself with the exact same magnitude of force. Wouldn't this also make the Earth move in a circle? Essentially, it does. The only thing is that Earth's mass is 81 times greater than the mass of the moon. This means that although it moves in a circle, it moves in a much smaller circle. The circle that the Earth moves around is so small that the center of this circle is inside the Earth. Cool, isn't it?
Sources:
Wikipedia
NASA
Rhett Allain's article in The Wired,
https://www.wired.com/2012/11/why-doesnt-the-moon-crash-into-the-earth/