These first tennis games were played with a variety of types of ball as well. 'Real tennis' used a cork ball that was wrapped tightly with wool fabric. (ref 1)
It is believed that lawn tennis took hold sometime in the 1870s and the ball then changed at that time. Between roughly 1480 and 1870 'real tennis' was played exclusively.
In Scotland, the early tennis balls were made from sheep's stomach (go figure) and sometimes goat stomachs. They were wrapped in wool and tied up with rope. Other Scottish tennis balls have been found that were made from animal fur, pine wood and innards (intestines and muscle) of various animals. In England early tennis balls were found to be made from human hair and putty. In Germany it is thought that the early tennis balls were in fact made from a rubber compound.
After the discovery of vulcanization, an Englishman named John Moyer Heathcote suggested covering these new rubberized tennis balls with Melton and the modern day tennis ball was born.
Today the International Tennis Foundation dictates the size and weight of the tennis ball. They are to be between 2.575 add 2.700 inches in diameter and must weigh between 1.975 and 2.095 ounces. The metric sizes are 65.41-68.58 mm and 56.0 grams to 59.4 grams respectively. (ref 2)
The modern manufacture process involves created 2 half-spheres of rubber. The two rubber halves have certain chemicals add inside and then sealed. The process creates a gas inside the now sealed ball that has a certain pressure. This gives the ball the correct bounce characteristics. The outside of the ball is textured and then, 2 pieces of cloth (either Melton or are needle cloth) are added to create the cover. The balls go through a final stage of tumbling in a steam environment to help the cloth fluff properly. And in the end you get your bouncy new tennis ball.
The ITF link above also includes a long list of approved tennis ball manufacturers (over 180 in total) sorted by country. It is also worthy to note that there are 3 kinds of tennis balls (stage 1, stage 2 and stage 3). The different stages are signified by the color of the ball. Stage 1 is green and is the most responsive ball. Stage 2 is orange and stage 3 is red. The orange and red balls are designed to allow for children and less skilled players to use a ball that is not as bouncy or fast which results in greater enjoyment of the game at lower skill levels.
There you have it. The long history of the tennis ball in one page. The next time you pick up a ball to play think of the hundreds of years of improvement the game has undergone and be thankful you are not holding a sheep stomach!
