Carbon fiber
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Carbon fibre can refer to carbon filament thread, or to felt or woven cloth made from carbon filaments. By extension, it is also used informally to mean any composite material made with carbon filament: for more on that application, see graphite-reinforced plastic. Aside from structural uses, carbon fibre cloth can be used in air filtration and as a high-surface area, corrosion-proof electrode.
Synthesis
Each carbon filament is made out of long, thin sheets of carbon similar to graphite. A common method of making carbon filaments is the oxidation and thermal pyrolysis of polyacrylonitrile (PAN), a polymer used in the creation of many synthetic materials. Like all polymers, polyacrylonitrile molecules are long chains, which are aligned in the process of drawing fibres. When heated in the correct fashion (http://www.psrc.usm.edu/macrog/carfsyn.htm), these chains bond side-to-side, forming narrow graphene sheets which eventually merge to form a single, jelly roll-shaped filament. The result is usually 93-95% carbon. Lower-quality fibre can be manufactured using pitch or rayon as the precursor instead of PAN.
Textile
These filaments are stranded into a thread. Carbon fibre thread is rated by the number of filaments per thread, in thousands. 3K (3,000 filament) carbon fibre is 3 times as strong as 1K carbon fibre, but is also 3 times as heavy. This thread can then be used to weave a carbon fibre cloth. The appearance of this cloth generally depends on the size of thread and the weave chosen. Carbon fibre is naturally a glossy black, but recently colored carbon fibre has become available.