Each person has the same set of genes - about 20,000 in all. The differences between people come from slight variations in these genes. For example, a person with red hair doesn't have the "red hair gene" while a person with brown hair has the "brown hair gene." Instead, all people have genes for hair color, and different versions of these genes dictate whether someone will be a redhead or a brunette.
One chromosome pair - the sex chromosomes - is unique. You don't necessarily end up with a matching pair. Typically females have two X chromosomes and males have an X and a Y. Mothers always pass an X chromosome on to their children. Whether your father passes on his X chromosome (leading to a pair of X chromosomes) or his Y chromosome (making a mixed set) determines your sex.
Though most adult cells contain two sets of chromosomes, sperm and egg cells are different. These special cells have just one chromosome from each pair. Which chromosome they get from each pair is random, making each sperm or egg cell unique. There is also a bit of mixing before the chromosomes are sorted into individual sperm or egg cells. Chromosomes from each pair in a mother or father, respectively, make contact and exchange pieces of DNA, creating hybrid chromosomes.
When a sperm and egg cell join together at fertilization they create a single cell with two complete sets of 23 chromosomes. This single cell divides to create new cells, over and over, forming the body of developing child. This is how you came to be you. If you have siblings, the same events created them. But because the egg and sperm production processes are random, your siblings didn't get the same set of chromosomes from each of your parents that you did (unless you happen to be an identical twin).
Because the chromosomes mothers and fathers pass on to their children are reshuffled versions of the ones they inherited from their own parents, it is hard to use most chromosomes to trace genealogy back very far. Most of the Y chromosome, however, is handed down from father to son without changes. Likewise, in humans, the tiny bit of DNA contained in an unusual package of genetic material known as mitochondrial DNA is passed down from mother to child without any recombination. For this reason, these types of DNA can be used to trace your ancestry.