'Normal' twins, (i.e. fraternal twins) are the result of multiple eggs in the womb, each one being fertilized separately. They are siblings sharing the same womb, and may be a male and a female, or be different in any other way two siblings can be different from each other. They may even have two different fathers.
Identical twins (or triplets, or more*) are the result of one egg splitting in two (or more) after it has been fertilized. Thus, identical twins both come from one original egg, and one original sperm. Each half develops into a separate, but genetically identical, being. Both twins will be of the same sex, and be nearly indistinguishable from each other.
Identical twins can share the womb with other, non-identical, babies including other sets of twins, triplets, etc., or even other sets of identicals. This is uncommon only because humans don't usually bear that many kids at one time, but when humans do have big broods, it is not uncommon to have sets of identicals in the mix.
Depending on the time that the egg splits, each baby may get its own placenta, chorion, and amniotic sac, or they may share. The great majority of identical twins share one chorion and placenta, but have two amnions (Monochorionic Diamniotic twins). Other possibilities are Dichorionic Diamniotic twins (in which the zygote splits before the sac is formed, and each twin has its own set of sac/placenta), and Monoamniotic Monochorionic twins (the zygotes split after the sacs and placenta form, and share the sacs and placenta**. MM twins are at risk for cord tangling and Twin to Twin Transfusion Syndrome TTTS).
Worldwide, identical twins occur in about 3 to 4 out of 1,000 births.
Even though they share 100% of their genes, identical twins have different finger prints and teeth marks. Social scientists have fun looking for other differences between identical twins, particularly those twins that were separated at birth, in order to determine which traits are influenced to what degree by environment and upbringing. The rest of us just think it's cool that they look exactly alike.
Identical twins are also known as monozygotic twins. Identical twins can also be mirror twins and/or conjoined twins (AKA Siamese twins).
See also: fraternal twins and half identical twins (AKA polar body twinning).
*The Dionne Quintuplets born in Canada in 1934 were identical quintuplets.
http://www.gen.umn.edu/faculty_staff/jensen/1135/example_student_projects/Sum2000/Twins/twins.html has a good chart illustrating this.