The Earth is always being pulled towards the Sun by gravity. If the Earth were stationary compared to the Sun, it would fall into the sun under the force of gravity. However the Earth is actually moving sideways compared to the center of the Sun at 3 km/second (~2 miles/second). The Earth is not moving fast enough to "escape" the Sun's gravity and leave the solar system, but it is going too fast to be pulled into the Sun. Therefore, it keeps going around and around - orbiting the Sun. It is rather like a tether ball. Think of the top of the post as the Sun and the ball as the Earth. The string between them is like the force of gravity keeping them the same distance apart. When you hit the tether ball it spins around the post. If there were no air or rope friction, the ball would spin forever without getting any closer to the post. That is essentially what the Earth is doing when it orbits the Sun - in the vacuum of outer space, it does not loose speed to air friction, so it just keeps going around the Sun.

Well, that's a good question, and Newton worried about the same thing! Actually, due to conservation of angular momentum, all the planets are in fairly stable orbits, with minor changes over millions of years, but no chance that they will fly off or anything! The earth's orbit is in the shape of an ellipse, which means that we get a little bit closer and farther from the sun over the course of a year. We also wobble in the tilt of our axis, so that the North Pole does not always point to the star Polaris, which is currently our north star.

But, the orbits are pretty stable, because there is a fairly constant gravitational force between the sun and the earth keeping the earth in its orbit. The strength of this force changes slightly over the course of an orbit, being a bit stronger when the earth is a bit closer - at those times(currently, when the northern hemisphere is having winter) the earth actually orbits a bit faster. (Not to be confused with spin!)

The Earth is "falling" around the Sun. The Earth has some initial momentum - it is moving in a direction, which is perpendicular to the direction of the Sun from the Earth. The Sun's gravity is enough to keep the Earth from flying off in a straight line, away from the Sun, but not enough to bring the Earth closer in - the Earth is continually changing its direction of movement, but in such a way that it follows a nearly circular path around the Sun.

If the Sun's gravity were stronger, it would pull the Earth in closer, but then the angle between the Earth's motion would also be changing more rapidly, so it would continue revolving around the Sun.

This concept is called the conservation of angular momentum, which is one of the basic principles of physics.

Kepler's law and Newtons laws explains this very well. If we could just STOP the Earth for a moment relative to the sun and then allow it to freely move it WOULD fall into the sun. But the earth was born from a ring of material that was MOVING around the sun on a stable orbit. So after the debris coagulated to form the earth, this initial orbital energy was retained. Hence the earth is moving on a stable orbit of fixed radius. Look up Kepler's laws, the period of revolution of a body squared is proportional to the distance between the sun and the body cubed.