It turns out that there is actually gravity in space. Its just a lot less than what we’ve got here on Earth (which makes us return straight to the ground after we jump!). The gravity in space does some important jobs – its what keeps the planets orbiting around the sun. Without it, they’d just float away!
The reason that gravity doesn’t really seem to work, like when we see the astronauts just floating around, is because what they’re experiencing is actually weightlessness (which is a bit different). This is because they’re basically in a free fall, in their spaceship as it orbits around the Earth. And there is gravity on the moon as well…its just less than there is on Earth.
The reason that there isn’t really much air in space is because space is just so big, the air isn’t concentrated like it is here on Earth, since out atmosphere keeps it in.
The earth’s gravity keeps the atmosphere and the air it contains close to the surface of our planet for the animals and plants to use. As eileen points out, in space, gravity is very very much weaker because there is very little matter (rocks, metals and gasses) to create any gravitational force so the there is no dense collection of air that we would breathe.