To make new cells, an existing cell divides in two. But first it copies its DNA so the new cells will each have a complete set of genetic instructions. Cells sometimes make mistakes during the copying process - kind of like typos. These typos lead to variations in the DNA sequence at particular locations, called single nucleotide polymorphisms, or SNPs (pronounced "snips").
SNPs can generate biological variation between people by causing differences in the recipes for proteins that are written in genes. Those differences can in turn influence a variety of traits such as appearance, disease susceptibility or response to drugs. While some SNPs lead to differences in health or physical appearance, most SNPs seem to lead to no observable differences between people at all.
DNA is passed from parent to child, so you inherit your SNPs versions from your parents. You will be a match with your siblings, grandparents, aunts, uncles, and cousins at many of these SNPs. But you will have far fewer matches with people to whom you are only distantly related. The number of SNPs where you match another person can therefore be used to tell how closely related you are.