It happens at the cellular stage. When we are still dealing with a zygote, this occurs within the first two days after the egg has been fertilised in about a quarter of monozygotic twins. The zygote splits into two and starts to form into two separate embryos. These twins are known as dichorionic diamniotic monozygotic twins, as they have separate placentas. The majority of the other twins split around day 3-4 and this will result in twins which share a placenta but are contained within their own sacs still, monochorionic diamniotic monozygotic twins. On very very rare occasions the split will happen much later, but the outcome is poor for those that do as it results in conjoined twins.
The reason some people don't realise there is more than one baby under formation in there is because they are so tiny at that stage that it is often only at the point that the heart starts to beat that it is picked up on a scan. By 8 weeks this will definitely be visible.
Dizygotic, or fraternal twins, occur from two separate eggs being fertilised.
Hope that helps explain it all for you.