The first step in tea production is the harvest. Most harvesting is still done by hand, which (as you can imagine) is very labor-intensive. Some growers have had success using a machine that acts much like a vacuum cleaner, sucking the leaves off the branch.
The latter method is used for the cheaper varieties of tea, as it is not capable of discriminating between the high-quality tip leaves and the coarser leaves toward the bottom of the branch.
The harvested leaves can be processed in two ways: CTC or orthodox.
CTC, which stands for "crush, tear, curl," is used primarily for lower-quality leaves. CTC processing is done by machine; its name is actually fairly descriptive. The machines rapidly compress withered tea leaves, forcing out most of their sap; they then tear the leaves and curl them tightly into balls that look something like instant coffee crystals. The leaves are then "fired," or dehydrated.
Most tea connoisseurs are not very interested in CTC tea, since this process does not allow for the careful treatment that high-quality leaves merit. But CTC has an important and legitimate role in the tea industry: since it is a mechanized process, it allows for the rapid processing of a high volume of leaves which otherwise would go to waste. It is also good for producing a strong, robust flavor from leaves of middling quality; in fact, for many varieties of leaf CTC is the preferred processing method.
The orthodox method is a bit more complex, and is usually done mostly by hand. The process differs for black, green, and oolong teas. The basic steps in the production of black tea are withering, rolling, oxidation, and firing.
First, the leaves are spread out in the open (preferably in the shade) until they wither and become limp. This is so that they can be rolled without breaking.
Rolling is the next step. This is rarely done by hand any more; it is more often done by machine. Rolling helps mix together a variety of chemicals found naturally within the leaves, enhancing oxidation. After rolling, the clumped leaves are broken up and set to oxidize.
Oxidation, which starts during rolling, is allowed to proceed for an amount of time that depends on the variety of leaf. Longer oxidation usually produces a less flavorful but more pungent tea. Many texts refer to the oxidation process by the misleading term "fermentation." However traditional and evocative the term may be, I think it is best avoided. Oxidation of tea leaves is a purely chemical process and has nothing to do with the yeast-based fermentation that produces bread or beer.
Finally, the leaves are heated, or "fired," to end the oxidation process and dehydrate them so that they can be stored.
Oolong is produced just like black tea, except that the leaves are oxidized for less time.
Green tea is not oxidized at all. Some varieties are not even withered, but are simply harvested, fired, and shipped out.