Outer space is the space which lies outside of the Earth's atmosphere (and, in the same way, all other planets' atmospheres). Of course, this is a rather simplistic definition: It incorporates some popular myths, and overlooks a few minor nuances.
DEFINITION OF OUTER SPACE
Essentially, outer space is the area which begins at the edge of Earth's atmosphere, and continues outward from there, essentially a vacuum - an area that is empty of all matter. Of course, that definition of "outer" space refers only to what is outside Earth's atmosphere. However, the area characterized by near-vacuum conditions, and the absence of air and gravity, obviously isn't just "everything away from Earth." Other planets (and some moons) in our solar system, for example, also have atmospheres. Technically, the near-vacuum of outer space also ends at the edges of their atmospheres, just as it does ours.
There is no clearcut line separating atmosphere from outer space, to be traversed by astronauts and rockets on a regular basis. Instead, the Earth's atmosphere peters out over a range of several miles, until it is non-detectable or non-existent. Formally, outer space is said to begin at the Karman line, about 62 miles above the surface of the Earth, at which the atmospheric pressure has dropped so far that it can no longer sustain any aircraft in normal flight. Alternatively, the U.S. Air Force has adopted a definition of space as everything above 50 miles.
OUTER SPACE IS NOT EMPTY
In the same way, there are differences in "outer space" within our solar system, and "outer space" outside of it. Within the solar system, the "solar wind," the stream of radiation and ions emitted by the sun, is a small but measurable force - so much so that, at least in theory, a ship could "sail" away from the Sun by using the energy from this background radiation. There is also a higher quantity of matter - basically, just free-floating hydrogen - floating around in the solar system as well, gases which escaped from the planets as well as the remnants of the giant molecular cloud which, billions of years ago, collapsed to form the Sun. Outside of the solar system, things get a lot colder and a lot emptier.
Even there, however, outer space is not actually empty space. We often refer to space as a "vacuum," a place that is empty of all matter. In terms of how humans relate to it, that may as well be correct: There is not enough air in space to breathe or to maintain a liveable air pressure, except with the aid of advanced technology (like spacecraft and space suits). However, space is not truly empty. The density of space is extraordinarily low - just a few atoms per metre, usually hydrogen and sometimes helium - but it is not empty.
Current theoretical physics actually suggests that outer space is even fuller, if only we could detect some of the theoretical forms of energy and matter which exist there, like dark matter. However, in terms of the confirmed and observable universe, outer space is the area outside of our atmosphere which is more or less empty except for small amounts of free-floating hydrogen.