Whether or not your twins are identical will depend on whether they are the result of a single fertilised egg splitting to form two babies (monozygotic), or whether they are the result of two separate eggs being fertilised by two separate sperm (dizygotic).
Identical/monozygotic twins can be further broken down into twins with separate placentas, twins that share a placenta, and even conjoined (or Siamese) twins depending on what stage in the conception process the egg was separated. If your twins share a placenta, they are definitely monozygotic twins. If they have separate placentas it is impossible to tell whether they are monozygotic or dizygotic until after birth. If you know you are carrying twins, early in your pregnancy you will have what is called a "chorionicity ultrasound' to determine whether your twins have one or two placentas. It is important to know whether your twins share a placenta or not as it will affect the way your antenatal care is handled.
Monochorionic twins are twins that share a placenta. They are the result of a single egg splitting, are identical and are always the same sex.
Monochorionic twins can be either Monochorionic-Monoamniotic which means that they share both a placenta and a single amniotic sac, or they can be Monochorionic-Diamniotic which means that they share a placenta, but that there is a dividing membrane between them, so they are in separate amniotic sacs.
All Monochorionic twin pregnancies carry around a 1 in 5 chance of developing Twin to Twin Transfusion Syndrome (TTTS) which is where the placenta shunts more blood to one twin than to the other resulting in a imbalance that can be fatal to both babies. For this reason, if you are pregnant with Monochorionic twins you will have regular growth scans from around 12-14 weeks to check that the babies growth is concordant.
With Monochorionic-Monoamniotic twins there is also a risk that the babies will become entangled in each others umbilical cords. This is not a risk with Monochorionic-Diamniotic twins as the dividing membrane prevents them from becoming entangled with each other.
Conjoined twins will always be Monochorionic-Monoamniotic.
Dichorionic twins are twins with separate placentas. They can be either the result of a single egg splitting (monozygotic), or they can be the result of two separate fertilised eggs (dizygotic). Dichorionic twins are always diamniotic, that is they are always in separate amniotic sacs. They are considered lower risk during pregnancy as there is no risk of TTTS developing, or of entanglement as they are in separate sacs. If your twins are different sexes, they are definitely the result of two separate fertilised eggs. If they are the same sex it is impossible to tell until after birth whether they are monozygotic or dizygotic. Unless they appear very different at birth, the only way to accurately determine zygosity is to have DNA testing carried out.