Chain mail (or mail or maille for the language purists) is a type of armor (or sometimes jewelry) made of interlocking links of wire.
Chain mail is generally more comfortable than any kind of interlocking plate armor, and is excellent defense against slashing and thrusting weapons like swords or arrows. Because it is next to useless when struck by bludgeoning weapons such as axes or maces, it is almost always worn over a padded undergarment called a gambeson (which, incidentally, also keeps one's body hair from being pulled out.) A common technique was to cover large inflexible areas of the body with plates, and cover joints with chain mail.
Though there are many different ways of interlocking the links to form a sheet, the most common one seen today in historical reenactment groups is called the European four-in-one, where each link is looped through the one above it, below it, and to the left and right. European six-in-one and European eight-in-one are similar. Other popular pattern families are the Japanese or Oriental weaves for sheets of chain, and the Persian family for making long, decorative chains. ASCII diagrams are woefully inadequate for explaining patterns, so I won't even try.
Both historical and modern mail has been made, almost exclusively, out of wire that is wrapped around a mandrel (dowel) to form a springy coil, and then cut into links. Links are then interlocked in the appropriate pattern, and bent into a closed loop. Due to a lack of manpower, most modern mail construction ends at this point, but archaeological finds of this "butted" mail armor are limited to obvious field repairs. In order to increase mail strength and durability, and to allow the use of smaller gauge wire, historical mail links were joined by riveting or, less often, welding. (Solid stamped "washers" were also sometimes used for up to half the links in a piece.) Links were riveted closed by overlapping the two ends of the wire in the link, pounding them flat, drilling a hole, then pounding a very short piece of wire into the hole. Mail-making required an enormous amount of labor, and in many areas was only feasible through the use of slaves.
As far as I am aware nobody has yet invented a chain mail making machine; modern mail is mostly made by bored SCA members while sitting in front of the TV. Around 2003 or so several wholesale mail-making factories opened up in India to cater to the first-world reenactor market, and while their wares are (relatively) cheap, the situation is a little too close to being "historically accurate" for my taste. A quick Google search will turn up vendors willing to sell you ready-made rings, and patterns are likewise easy to find.
For a particularly stunning use of chain mail armor, see lorica plumata.