What do you mean by "stay up?" If you mean, stay in orbit around the sun, that is due to the gravitational pull of the sun keeping the earth orbiting around it.

In "space" there is no up or down; up and down are concepts that humans developed on the earth because we fall toward the center of the earth. Since the earth is really huge compared to us, we perceive the earth to be flat, and our feet are "down" and our head is "up" - that is, toward the sky. But the sky looks up to everyone, no matter where on the earth you are, whether in North America or Australia.

You know the earth is round, right? So, why is the earth round is actually an interesting question. It is because a sphere (that is the scientific term for a round ball shape) is the shape that takes the least amount of energy to form and to maintain! The earth was once a hot ball of molten rock, and it used to spin even faster than it does today. It assumed the shape of a sphere as it was forming. The shape that is most stable for liquids in space is a sphere. In the space shuttle, or nowadays, the orbiting space station drops of liquid are always round! Even flames in the shuttle (where there is oxygen!) assume a round shape. But - here's something really cool:
because the earth is spinning, it is not perfectly round. It bulges a bit at the equator! It is just a little wider at the equator than at the poles, so it is somewhat flattened, like an extremely fat (that's FAT, not flat) pancake!

I assume you are asking how does the Earth stay (roughly)the same distance that it is from the Sun. The answer is gravity and angular velocity - the gravitational pull of the Sun on the Earth keeps the Earth from flying away and the fact that the Earth has angular momentum keeps it rotating around and not falling in to the Sun. Imagine swinging a small weight on the end of a string in a circle. In this analogy, you are Sun, the string is gravity, and the Earth is the small weight. As long as the weight moves at the same speed and the string doesn't break, the weight will keep going around and around.

The Earth is round because of the way gravity works and that when the Earth formed it was made from stuff like dust, small rocks, and other particles. As gravity depends only on the distance between objects, when the stuff that the Earth was made of came together everything got as close together as possible with no preferred direction. So, the Earth became round. Why do you think some smaller objects such as small moons and asteroids can have interesting non-round shapes?