Kevlar's chemical structure naturally makes it form in tiny straight rods that pack closely together, like lots of stiff new pencils stuffed tightly into a box (only without the box). These rods form extra bonds between one another (known as hydrogen bonds) giving extra strength—as though you'd glued the pencils together as well. This bonded rod structure is essentially what gives Kevlar its amazing properties. (More technically speaking, we can say the Kevlar rods are showing what's called nematic behavior (lining up in the same direction), which is also what happens in the liquid crystals used in LCDs (liquid crystal displays).)
You probably know that natural materials such as wool and cotton have to be spun into fibers before they can turned into useful textile products—and the same is true of artificial fibers such as nylon, Kevlar, and Nomex. The basic aramid is turned into fibers by a process called wet spinning, which involves forcing a hot, concentrated, and very viscous solution of poly-para-phenylene terephthalamide through a spinneret (a metal former a bit like a sieve) to make long, thin, strong, and stiff fibers that are wound onto drums. The fibers are then cut to length and woven into a tough mat to make the super-strong, super-stiff finished material we know as Kevlar.