Let's just go back to the begining, just after fertilization, when a single cell called a zygote is formed, containing the 46 human chromosomes, half of which are from the mother's egg cell, the other half from the sperm of the father.

Normally this zygote cell will divide many times by mitosis to give rise to a ball of cells, which eventually gives rise to the embryo and all the tissues needed for its development during pregnancy.

So how are twins formed?

Monozygotic twins come from the same zygote - it's just that during the first few cell divisions, instead of staying together and forming one embryo, two or more cells separated and each of them gave rise to an embryo.  These two embryos therefore contain copies of the same parental chromosomes. (They may or may not share a single placenta)

Fraternal, or dizygotic twins were never the same cell.  They are formed when two different egg cells are fertilized by two different sperm, so they are like brothers and sisters, it's just that they happened to be fertilized at around the same time.

So, how do we explain the observation that monozygotic (identical) twins are not actually identical, since all their cells contain copies of the same DNA?

One possibility is that mutations (changes in the DNA itself, maybe due to mistakes when the DNA copies itself to make new cells) occurring over the lifetime could change development slightly and give the subtle differences we notice between identical twins.

But now another way has been shown, which enlightens the influence of the environment on developing cells.

Differences in the methylation (adding methyl groups) of DNA can change the expression of the DNA (turning a sequence of DNA, called a gene, OFF so that no protein is made)

The control over turning genes ON or OFF, called gene expression, is vital - since all the cells in our body contain copies of the same genes, some genes must be turned on in certain tissues, for example muscle cells, but turned off in other non-muscle tissues.  That's how we get different tissues doing different things in our body.

The DNA itself can control gene expression, but so can methylation of DNA.

Patterns of methylation of DNA are partly inherited from our parents, but the methylation pattern is mostly wiped out from the parents chromosomes and a new pattern is added, right from the stage of the zygote and during pregnancy.

So, if methylation turns genes OFF, and if methylation can change over our lifetime due to differences in the environment of the DNA, that is a way that identical twins can be different from eachother.

These effects of the environment on inheritance is known as epigenetics

Action:

Wanna see some cool dudes workin' on DNA methylation?  -  video here ya'all!