How do cats purr?
Strange but true, the actual mechanism remains a mystery.
Intuitively it would seem that that the sound is caused by air moving over the vocal cords, kind of like humming. But while most theories link purring to breathing, there is some disagreement about what is actually causing the vibration. Some say that the moving air vibrates the false vocal cords, which are 2 folds of membrane behind the real vocal cords. Another theory is that muscles attatched to the larynx contract, which causes a sort of fluttering of the glottis. As the cat breathes, the air gets pushed past the glottis, causing the sound. Or the sound might be produced in the hyoid apparatus, which is a series of small bones attached to the larynx and the tongue. Or the vibration is in a ligament connecting the clavicle bone to the throat.
Another, less popular theory is that purring is related to blood flow. Again, how this actually manifests itself is unclear. It could be increased blood flow or constriction of blood vessels in the throat or the soft palate in the mouth, causing vibrations which resonate in the throat. Or it could begin in the vena cava, which is the main vein going into the heart. This vein is constricted where it passes the liver and diaphragm, and sometimes the blood gets backed up, causing a vibration as it forces its way through. The vibration resonates in the cat's body, up to the throat and sinuses, causing the purring sound.
Almost all species of cat can purr. The exception seems to be the ones that can roar: lions, tigers, leopards, and jaguars. The vocalization apparatus is different in these species. There is discussion of this, with pictures and sounds at http://www.lam.mus.ca.us/cats/P14/alt.htm. If we are going with the "purring by air vibration" theory, apparently when large cats (e.g. mountain lions) do purr, they can only purr on the exhale, unlike domestic cats who can purr inhaling and exhaling. Some scientists believe that purring is linked to the central nervous system and is a completely voluntary action. That is, the purring mechanism is going all the time, but cats can adjust the volume as they want to. Several articles, including the writeup above, say that domestic cats purr at 27-44 hertz. Several other sites say that they purr at about 26 hertz, roughly the same frequency as a diesel engine. I have no idea which is true.