Let's get some concepts defined first, shall we?
Mitosis: how one (human) cell with 46 chromosomes becomes two cells with 46 chromosomes each [the DNA replicates, one copy goes to one side, other copy to the other side, and it splits down the middle]
Meiosis: how one (human) cell with 46 chromosomes becomes two sex cells, or gametes, with 23 chromosomes each [the cell skips the DNA replication]
Gametes: the biology textbook term for sex cells, with males producing spermatozoa (sperm) with either the X or Y sex chromosome [usually males make about 50% of each] throughout their lives and females born with ova (eggs) that are always X
Chromosome: organized piece of DNA (they are generally portrayed to look somewhat like an elongated "x")
When a mommy and a daddy love each other veeerrrry much... Just kidding. Let's skip to fertilization.
The lucky male gamete that first got through the outer layer of the female gamete has 23 chromosomes. This is convenient, because the female gamete has 23 gametes as well, and 23+23 = 46. Chromosomal abnormalities known as polysomy, the most well-known of which is probably Down Syndrome, are where this math is different.
Now that the chromosomes from Mom and Dad are mingling, all sorts of reactions are occurring within the one cell that used to be two. The egg/sperm combo is now a zygote (soon-to-be embryo), and it starts to develop. It doesn't begin to grow for the first week or so; rather, the cell undergoes mitosis and divides, or cleaves, into smaller and smaller cells all stuck together. This ball of dividing cells is known as a blastocyst.
Here's the kicker: the cleavage of the blastocyst can cut it right down the middle, leaving two bunches of cells. If the split was even and clean, everything is OK (for now*), and identical twins develop. Uneven splits make the embryos nonviable (as far as I know, let me know if I am mistaken). An embryo that splits late, or incompletely, produces conjoined twins.
The scientific term for identicals is "monozygotic twins" because they are produced from one zygote. Fraternals are "dizygotic twins."
See why a boy/girl pair cannot be identical? XX cannot split into XX and XY, just as XY wouldn't make XY and XX. For convenience, we are going to disregard further complexities from my little ol' high school bio explanation, such as sex chromosome abnormalities and semi-identical twins.
XO (or, in my chromosomal case, XX),
*Twin development in utero can have complications special to twins, like twin to twin tranfusion, but that's another post.