Blood types are inherited and represent genetic markers from both parents, so the answer to this question depends on the blood types of the parents. Each biological parent donates one of their two ABO alleles to their child. If both parents are Type A, for example, then their children must also all be Type A. But if parents have different blood types (any combination of A, B, and O) you can — and likely will — see variations among the blood types of their offspring.
Identical twins will always have the same blood type because they were created from the same fertilized egg (fraternal twins can have different blood types — again, providing the parents do — because they are created by two fertilized eggs).
It's possible for a pregnant woman to carry a fetus with a different blood type than her own, and sometimes the mother forms antibodies against the red blood cells of the fetus, leading to low fetal blood counts. This fairly common condition is known as hemolytic disease of the newborn.
Check out this Web site, which contains a calculator where you can see the possible combinations that could occur from parents and their blood types.