Just to clarify the above (quite correct) answers, for the sake of people who aren’t used to thinking in these terms:
Identical twins result from one egg + one sperm = 1 embryo, which splits at a very early stage (when all the embryonic cells are all pretty much the same, and haven’t decided their roles yet) into two (or more) parts. This early in the developmental process, each surviving part can “catch up” with just 1-2 cell divisions and develop into a separate, identical twin (or triplet)
If the “split” happened after the first cellular division (i.e. the two-cell stage becomes two one-cell stages), each embryo would behave as if it were completely independent. Such a pregnancy would develop exactly like a pair of fraternal twins: each embryo would implant separately, usually in different parts of the uterus, and will form it’s own placenta, amnionic sac, etc. (like two seeds from the same pod, taking root separately) Since the window of opportunity for this is pretty narrow, it’s not terribly common. I’m afraid I don’t recall the exact figure.
The later the split happens, the more pregnancy structures the two embryos may share. If the split occurs after the has embryo implanted in the uterine wall and begun to develop a chorion (the part that links the embryo to the mother) the two twins will share that, like two trunks from the same root.
If the split occurs after the chorion has begun to form, but before the amniotic sac (the sac around the embryo) has begun to develop in earnest, each embryo may grow its own amnion. The “two trunks” will be like independent trees. If the amnion has already begun to form, both embryos may form in one sac - literally “two peas in a pod”. This can be a cramped situattion, and the risk of tangling limbs and umbilici can make this a very high risk pregnancy.
If two zygotes happen to implant near each other, the two placentas may overlap as they grow, but it’s usually or often possible to tell that they began as two independent implantations. In such cases, we presume that the twins are fraternal (non-identical) but only genetic testing after birth can confirm this.
I hope this clarifies more than it confuses. It’s much easier to explain with a sketch.