Niels Bohr came up with a model of electrons orbiting nuclei, which was very similar to how planets orbit the Sun. The electrons were like the planets and the nucleus was like the Sun, but with some peculiar rules about which orbits were possible, very unlike any rules for planets.
In part because this theory didn't agree with a lot of how electrons really behave, quantum mechanics was developed. In quantum mechanics, the electron doesn't orbit the nucleus like a planet, but instead it forms a cloud around the nucleus. If you look (with special instruments) to see where the electron is, the cloud only tells you the odds of finding the electron in different places. Before you look, it's spread out! If that sounds weird to you, it sounds even weirder the more you think about it.
The quantum picture does have a little bit in common with the planetary picture. In both of them the negative potential energy due to the attractive force is twice as big as the positive kinetic energy.
The following picture is a rough illustration of an electron cloud, borrowed from the site named in the picture.
There are many possible shapes; the sphere shown is just the simplest. The little blow-up is supposed to convey some idea of nuclear structure. The electron is more likely to be found near the nucleus than far away so the cloud is drawn as darker near the nucleus. Thus the darker a point is, the higher is the electron's probability for being found there.
Thanks for the question!
~Ann (and Paul)
p.s. For an amusing thread on this question, you might want to look at http://van.physics.illinois.edu/qa/listing.php?id=1295
(published on 10/22/2007)