Nature is absolutely amazing when it comes to reproduction. There is a system specifically designed to make children different.
A person has a set of chromosomes. Each chromosome contains two halves that join in the middle so they look like an "X." All that a chromosome is, by the way, is two very tightly coiled strands of DNA (deoxyribonucleic acid).
The two halves of each chromosome come from the two parents. One half of the "X" in each chromosome comes from the mother, and the other half comes from the father. The two halves are bonded together at the middle of the "X." Each half of the "X" contains a complete set of genes, so each chromosome has two copies of every gene -- the "dominant" gene of the two is the one that is expressed.
A sperm and an egg meet to create a new person. The sperm carries one half of the new person's ultimate chromosomes, and the egg contains the other half.
The question then comes down to, "Where does the DNA in the sperm and the egg come from?" Every cell in the father contains a complete set of X-shaped chromosomes that are exactly alike. Same for the mother. The father's chromosomes came from his mother and father, and the mother's from her mother and father. To make a sperm cell, only one half of the X is contributed. But which half? This is where nature does the especially amazing part. When forming sperm cells, the father's body randomly chooses genes from the two halves of the father's chromosomes. This means that every sperm cell contains a random mix of the father's parents' genes. The same thing happens when forming eggs. Therefore, each child that a couple produces is a random mix of the four grandparents' genes.
Here are some interesting links: