I am amazed at this question because I have twins daughters in the 3rd grade at Roosevelt School--so it is funny that you asked me this question.

I am a geologist and an astronomer (an astrogeologist) and so I am not an expert on biology. But this is how I understand it works.

Twins that are called IDENTICAL form when the egg in the woman spontaneously splits into two when it first starts growing inside the woman's uterus. So in this case, a single egg splits into two and the twins that form are identical twins. Identical twins are always the same sex...2 boys or two girls, NEVER one of each, because there was originally only a single egg within the mom before it split into two.

The other kind of twins are called FRATERNAL TWINS. In this case the woman makes TWO INDEPENDENT eggs (called multiple ovulation) inside her body and each of these eggs grows and develops into a baby. The eggs were NEVER THE SAME. Hence fraternal twins can be the same sex or different sex; there is a 50 % chance that they are the same.

The difference between identical twins and other twins is in the first stages of fertilization.Usually, a woman produces a single egg, which, if fertilized by a man's sperm, can produce a single child. The woman's egg has 1/2 of the genetic material of the mother, and is joined with the sperm, which contains 1/2 of the genetic material of the father. But no two eggs or sperm are just the same. In making the sperm and the egg, the genes are chosen more or less at random from the possible choices of both mother's and father's genes. It's like flipping a coin to make a choice between two things - it is tough to predict what you will wind up with. By mixing together some of the genes of the mother and some of the father, we get a whole new genetically distinct person - the child. This is why everyone is different - or almost everyone except identical twins.

Twins can occur in two distinct ways.If a woman produces two eggs during a single cycle, both eggs can be fertilized by two different sperm. Each egg has a distinct genetic code, as does each sperm. So, while the two twins share the womb, they do not share the same genetic material, so they can be distinctly different. One can be a boy and the other a girl, or both can be boys or girls. It is just that they shared the mother's womb for a time that makes them twins.

For identical twins, it is quite different. Identical twins start out as one egg fertilized by a single sperm. All things grow by a process of cell division - one cell divides into two, two into four, etc. Early in this process of cell division, the cells that will make identical twins split into two distinct groups. Basically, what should have been one individual becomes two separate individuals. But because they came from the same egg and sperm, they have exactly the same genes. So, the two will develop identically and become identical twins. Identical twins are more like each other than any other human being is like any other human being.

The mixing of genes that occurs during fertilization is also one reason why everyone is so different. Each of us has a little bit different genetic material than everyone else. Genes are the instruction set for how we grow, how big, how much we weigh, what color skin and hair, and so on. This mixing of genes occurs every generation, and is what makes living things so adaptable.

The genetic code for a female is XX, the genetic code for a male is XY. The sex of a baby is determined by the father's sperm. All eggs inside a mother are "X". The father's sperm is either X or Y. When the egg and sperm combine they create either a boy XY, or a girl XX. So within the father, there are millions and millions of sperm, some X, some Y. So any father can produce as many boys or girls as need be. It all depends on which type of sperm gets to the egg first, an X sperm or a Y sperm, that determines whether the baby will be a boy or a girl.
When twin babies develop, it can be via one of two ways. Either, the mom can produce 2 eggs at once which each get fertilized by a sperm. These are called fraternal twins. Since they are two totally separate eggs and separate sperm, they can be either one boy, one girl, or both girls, or both boys. Since these twins arise from separate eggs and separate sperm, each baby is genetically different from each other. It's just like having a double pregnancy.
The other way twins can develop is by a single egg getting fertilized and then as that embryo begins to develop, it spontaneously splits into two totally separate embryos and they both develop simultaneously each as a whole baby. Since in this case it is all begun with one egg and one sperm, these types of twins will be always the same sex, either both boys or both girls, depending on whether an X or a Y sperm fertilized the egg. These twins would be known as identical twins, because they have the exact same genetic makeup.
What about triplets or quadruplet babies? Could you have identical and fraternal twins in each of these cases?

There are 2 ways to get twins, which is why there are 2 kinds of twins:
identical and 'fraternal.'

Fraternal twins happen when 2 different embryos start growing at the same time, so you get a brother and a sister or 2 sisters or 2 brothers that are the same age.

Identical twins happen when 1 embryo splits into 2 embryos when it is only a few cells big, so both babies have the same genes.

I guess triplets and quadruplets and more-lets probably aren't usually identical, because it's hard to imagine an embryo separating into more than 2 pieces that can each develop into a baby. I wonder what is the largest number of identical babies that's ever been born from a woman. Do you think there are ever identical triplets? identical quadruplets? Maybe identical quadruplets are more common than identical triplets, because the way the embryo grows is that 2 cells divide into 4 cells, which divide into 8 cells, so it seems like embryos would be more likely to split into halves and quarters than into thirds.

I wonder how many cells big the embryo is when it splits into twins. I'd guess most embryos split into twins when they are 2 cells big, and some when they are 4 cells big, and a few when they are 8 cells big, but that is only a guess.

When the embryo develops, it starts out as a ball of cells. Then the cells move so that they're all on the surface and the embryo is like a hollow ball. Then the ball collapses and sort of folds in half. That's when the exciting changes take place, and cells on different parts of the embryo turn into different things like nerves and skin and gut.

Good luck with your science, and let me know if you have any other questions.