An electron bound to the nucleus of an atom is often thought of as orbiting the nucleus in much the same manner that a planet orbits a sun, but this is not a valid visualization. An electron is not bound by gravity, but by the Coulomb force, whose direction depends on the sign of the particles' charge. (Remember, opposites attract, so the negative electron is attracted to the positive proton in the nucleus.) Although both the Coulomb force and the gravitational force depend inversely on the square of the distance between the objects of interest, and both are central forces, there are important differences. In the classical picture, an accelerating charged particle, like the electron (a circling body changes direction, so it is always accelerating) should radiate and lose energy, and therefore spiral in towards the nucleus of an atom... but it does not.
Since we are discussing a very small (microscopic) system, an electron must be described using quantum mechanical rules rather than the classical rules which govern planetary motion. According to quantum mechanics, an electron can be a wave or a particle, depending on what kind of measurement one makes. Because of its wave nature, one can never predict where in its orbit around the nucleus an electron will be found. One can only calculate whether there is a high probability that it will be located at certain points when a measurement is made.
The electron is therefore described in terms of its probability distribution or probability density. This probability distribution does not have definite cutoff points; its edges are somewhat fuzzy. Hence the term "electron cloud." This cloudy probability distribution takes on different shapes, depending on the state of the atom. At room temperature, most atoms exist in their lowest energy state or "ground" state. If energy is added-by shooting a laser at it, for example-the outer electrons can "jump" to a higher state (think larger orbit, if it helps). According to quantum mechanical rules, there are only certain specific states to which an electron can jump. These discrete states are labeled by quantum numbers. The letters designating the basic quantum numbers are n, l, and m, where n is the principal or energy quantum number, l relates to the orbital angular momentum of the electron, and m is a magnetic quantum number. The principal quantum number n can take integer values from 1 to infinity. For the same electron, l can be any integer from 0 to (n - 1), and m can have any integer value from -l to l. For example, if n = 3, we can have states with l= 2, 1, or 0. For the state with n = 3 and l = 2, we could have m = -2, -1, 0, 1, or 2.
Each set of n, l, m quantum numbers describes a different probability distribution for the electron. A larger n means the electron is most likely to be found farther from the nucleus. For n = 1, l and m must be 0, and the electron cloud is spherical about the nucleus. For n = 2, l = 0, there are two concentric spherical shells of probability about the nucleus. For n = 2, l = 1, the cloud is more barbell-shaped. We can even have a daisy shape when l = 3. The distributions can become quite complicated.
Experiment has verified these distributions for one-electron atoms, but the wave function computations can be very difficult for atoms with more than one electron in their outer shell. In fact, when the motion of more than one electron is taken into account, it can take days for the largest computer to output probability distributions for even a low-lying state, and simplifying approximations must often be made.
Overall, however, the quantum mechanical wave equation, as developed by Schrödinger in 1926, gives an excellent description of how the microscopic world is observed to behave, and we must admit that while quantum mechanics may not be precise, it is accurate.