Birth happens during the last couple of hours of pregnancy, your pregnant Sim will not be able to do anything, and will automatically head home. When the baby starts coming, the mother will grasp her stomach and moan, and give birth after a couple of hours. She will also gain a positive moodlet called Baby Is Coming, but it doesn't give any Focus boost.
In The Sims Medieval, there is no hospital, so the mother will have to have a home birth. The player does not get to choose the baby's traits or fatal flaws. After giving birth, the mother will get It's a Boy/Girl, depending on the gender of the baby. Every baby is born with a pre-chosen bassinet, dependent on the parent's Hero profession.