Modern contact lenses fall into two categories: soft lenses that are made from water-containing plastics, and GP or "oxygen permeable" rigid contact lenses.
Contact lenses may also be classified by wearing schedule. Daily wear lenses must be removed, cleaned and stored each night, while extended wear contact lenses are made from materials which are safe for sleep. You may also have heard of "continuous wear" contact lenses, a type of extended wear that can be worn for up to 30 days.
Various lens designs are available for different vision problems. Spherical contact lenses correct nearsightedness or farsightedness and are indicated by a minus or plus in your prescription, respectively. Bifocal contact lenses are similar to multifocal eyeglasses in that they use different optical zones to correct presbyopia (the decreased ability see at both near and far distances). Toric contact lenses correct astigmatism, which can accompany either nearsightedness or farsightedness.
All of the contact lenses mentioned above can be custom fabricated for unusual prescriptions, and many other contact lens designs are available as well, including designs for conditions like keratoconus.