You can get pregnant before your period, that is the time it is going to happen because a woman will ovulate sometime after her period. If you get pregnant then you will not have a "normal" period--because a normal period would be you getting rid of everything and your body will not do that when it is getting your body ready to carry a baby. But what can happen is to have bleeding during that time which can be mistaken for a period, sometimes it doesnt last as long or is light compared to a normal period.
If a woman has sex one day and her regular period begins the next day (or even in the next few days) she is not pregnant.