1) How does the Earth revolve around the sun?
Here, you have to remember that the Earth is moving at great speeds. It is this movement that keeps the Earth from falling into the sun. Now recall Newton's first law of motion: inertia, a body in motion tends to stay in motion. Because of inertia, the Earth would like to move in a straight line. However, due to the gravitational force between the Earth and the sun, the Earth is constantly "falling" toward the sun. This gravitational force curves the Earth's preferred straight line of motion into a circle. You can think of this as the sideways motion of the Earth keeping it from ever reaching the sun (see
http://en.wikipedia.org/wiki/Orbit#Understanding_orbits for a better explanation).
Early models of the atom were based on this "planetary orbit" model. Electrons travel very fast and their fast speeds allow them to orbit the nucleus like planets orbit the sun.
2) Why is this model is incorrect?
Now, if you have studied electricity and magnetism, you should see a problem with this planetary model. The electron is traveling in a circle which means that it is constantly accelerating (note: although it is not changing speed, it is changing velocity because velocity accounts for the direction of the electron as well as its speed), and the electron has a charge. From the laws of electricity and magnetism, we know that accelerating charged particles should radiate energy (in the form of electromagnetic radiation). Because energy is conserved, this radiated energy would decrease the speed of the electron (lowering its kinetic energy), causing it to slowly spiral into the nucleus. So, we're back where we started, classical physics predicts that an electron will fall into the nucleus!
3) WTF?! What's really going on?
The real explanation for why an electron does not fall into the nucleus comes from a fundamental concept in quantum mechanics: the Heisenberg uncertainty principle. Put simply, it states that you cannot know the position and momentum of a particle simultaneously. More rigorously stated, the product of the uncertainty of the position of a particle (Δx) and the uncertainty of its momentum (Δp) must be greater than a specified value:
$$\Delta x \Delta p \geq \frac{\hbar}{2}$$
Now, as the electron approaches the nucleus, it's uncertainty in position decreases (if the electron is 10nm away from the nucleus, it could be anywhere within a spherical shell of radius 10nm, but if the electron is only 0.1nm away from the nucleus, that area is greatly reduced). According to the Heisenberg uncertainty principle, if you decrease the uncertainty of the electrons position, the uncertainty in its momentum must increase. This increased momentum uncertainty means that the electron will be moving away from the nucleus faster, on average.
Put another way, if we do know that at one instant, that the electron is right on top of the nucleus, we lose all information about where the electron will be at the next instant. It could stay at the nucleus, it could be slightly to the left or to the right, or it could very likely be very far away from the nucleus. Therefore, because of the the uncertainty principle it is impossible for the electron to fall into the nucleus and stay in the nucleus.