What is purring?
Purring is a sound all cat lovers have come to know and enjoy. It is not entirely understood how cats purr but it is believed to be the result of rhythmic contractions of the muscles of the voice box and diaphragm which creates pressure effects that result in turbulent airflow through the trachea. 
Domestic cats purr in a frequency of 25 to 150 Hz per second. Purring isn’t confined to domesticated cats only, some species of wild cats such as cheetahs, bobcat, serval, ocelot, and the lynx.
Why do cats purr?
There are many reasons why cats purr. New kittens are able to purr as early as two days old.
The purr of a kitten has a higher pitch than other types of purring. Kittens purr when they are nursing and their mother will purr back, it is believed to be a signal to the mother that all is well It is believed that this purring elicits a care-giving hormonal response.
Purring in adult cats
Purring isn’t confined to kittens, it continues into adulthood. Pet owners usually associate purring as a sign of a contented cat, however, purring is also known to occur in a distressed, sick or even dying cat.
In this situation, it may be a throwback to when cats lived in the wild, a weak or dying cat is vulnerable to predators, therefore purring may help to throw off any potential hunters by suggesting that all is well with the cat.
Female cats will often purr when they are giving birth. It is believed that when cats are under stress they may purr as a way to calm themselves down and or as a sign of submissiveness when afraid (for example when they are at the veterinarian).
Interestingly, it is also believed that purring has a strengthening and healing effect on the bones, tissues and respiratory system.  As we know, cats purr at a frequency of between 25-150Hz per second, which is the range which can help improve bone density and promote healing.
 Cat Owner’s Home Veterinary Handbook – Delbert G. Carlson & James M. Griffin.