There is no air in space because the air on earth is made up of gas molecules that, believe it or not, have mass. That means they are subject to the force of gravity. Gravity is strongest when one is closest to a large body like the earth. The air at sea level (closest to the earth) is held in place by its own weight, plus the weight of all the air above it in the atmosphere. At the tops of high mountains, there is literally less air, fewer gas molecules in a given volume. This is because (1) there are fewer air molecules above it in the atmosphere and (2) the force of gravity is less at a larger distance from the earth. Outer space is so far away from the earth that the force of gravity is too small to hold the air molecules in place. You may then ask why there is no air on the moon, since it is a large body with gravity. This points to another cool feature of our earth: it also has a magnetic field. This magnetic field helps keep the air from being blasted away by solar wind. Solar wind is electromagnetic radiation from the sun that moves out into our solar system like wind, giving energy and electric charge to all the molecules it encounters. Because air molecules are so light, they can be "blown" away by this wind--and indeed they are blown away from the moon (a rocky body without a magnetic field), which is why it has no air. The earth is a solid rocky body with a core of moving liquid iron, which creates a magnetic field that deflects the solar wind and allows us to keep our air.

The quick answer is gravity, or the lack thereof. Air is composed of small molecules of gas, that while you cannot see, actually have mass, just like your body, or a ball, or any other object that you can see around you. First, think of your body, the ball, or other objects that you can see- they are all clearly sitting safely on the surface of the earth- not floating away into space. Why? Anything that has mass will be affected by the pull of gravity, and gravity is much stronger at the surface of planets than far away in space. So even though you cannot see them, the small gas molecules that make up the air behave just the same as the objects you can see and get stuck close to the surface of Earth and the other planets, and are not able to escape gravity and go into space.

First, there is (in some sense) air in space: outer space is not completely empty, but the density of "stuff" (dust particles, ions, gases, etc.) in it is far lower than in and around planets, stars, and other celestial bodies. However, the density is so low that, for practical purposes (like breathing), space can be considered empty. This emptiness is sort of due to the fact that there is so much space and so little stuff to put in it, which has to do with some fundamental constants that define the universe, the origin of whose values we don't know.

The lack of air between planets/stars/etc. also has to do with gravity. Air (or rather, an atmosphere) is held around a plane by gravity, which is the force that attracts all objects with mass toward each other. Molecules of the atmosphere have mass, Earth has mass, and gravity pulls them together.

During the initial formation of the planets, there was gas throughout the solar system. As the planets coalesced, gravity pulled the gas molecules toward them, forming the atmospheres and emptying the surrounding space. Atmospheres can be lost though. One way is related to temperature. As temperature increases, the speed of the molecules in a material also increases. Molecules can be moving fast enough to exceed the escape velocity of the planet and will then fly out into the emptiness of outer space. Another mechanism by which planets lose atmosphere is by "leaking" ions along magnetic field lines. This is happening to Earth and is responsible for a loss of about 1 kg of atmosphere per day. Other mechanisms include impact erosion, wherein impact by large meteoroids causes atmospheric molecules to reach the speeds necessary to escape, and interactions with the planet, such as being dissolved into liquid (like the oceans) or reacting with other elements (such as rusting, which combines oxygen from air with iron).

Advanced readers may be interested in this paper.

Also see here for a sort of interactive resource on retention and loss of atmospheres by planets.

Air, like everything else, is affected by gravity, so it falls onto planets and other things that have gravity. Air is a gas, though, so it doesn't cling to the surface the way that a water would, but it still sticks to planet due to their gravity.