To begin, we have to think about Schrödinger's Equation, which is the equation that determines how probability waves, such as, in this case, electrons in the nuclear potential, propagate. By solving this equation, we can find what are known as eigenstates of the energy. An energy eigenstate is a wavefunction that is "stationary," which is to say the probability density at any point doesn't change in time. This is to contrast with states that are not stationary, which are combinations of eigenstates. An electron can, in principle, be in any combination of these eigenstates. Anyway, I'll limit the discussion to the eigenstates of the electrons in the nuclear potential, which what give rise to our concept of atomic orbitals.
The eigenstates in the potential made by an atomic nucleus can be described by the quantum numbers n, l, and m. n is the "principal quantum number" and it largely determines the energy of an electron in the potential well of the nucleus. l is the "azimuthal quantum number" or sometimes called the "angular momentum quantum number" which describes the total angular momentum of the electron. We use l to divide energy levels into different "subshells" which are designated with s, p, d, f, etc. m is the "magnetic quantum number" and describes the angular momentum of the electron along a particular axis (usually, we use the z-axis, but this choice is ultimately arbitrary). Together, n, l and m describe the shape of the orbitals. Below is what I think is a good illustration of the variety of shapes of orbitals you can get with different quantum numbers (the first number is n, the second is l, and the third is m):
"The Physics of the Universe: Probability Waves and Complementarity." <http://www.physicsoftheuniverse.com/topics_quantum_probability.html>.
The purple parts of the orbitals denote areas of high electron probability, and the orange parts denote areas of low probability.
If you want to explore more the dynamics of the hydrogen orbitals, you can check out this applet: http://www.falstad.com/qmatom/
Hope that answers your question,
Falstad, Paul. "Math and Physics Applets." http://www.falstad.com/qmatom/
(published on 04/03/2011)