First cousins share a set of grandparents, second cousins share a set of great-grandparents, and third cousins share a set of great-great-grandparents. As the generations between the two cousins increases, so does the degree of cousinhood.
The degree of cousinhood (first, second or third) is determined by the number of generations between two cousins and their nearest common ancestor. First cousins are the children of a person's aunt or uncle. For instance, Polly's mother has a sister, Suzie. Suzie has a daughter named Lilly. Polly and Lilly are first cousins. Since Polly's mother and her sister Suzie have the same parents, Polly and Lilly share a set of grandparents. This means there is one generation between Polly and Lilly and their common ancestor.
Second cousins are the children of first cousins. Polly's children and Lilly's children would be second cousins since they share a set of great-grandparents. This means there are two generations between them and a common ancestor.
Third cousins are the grandchildren of first cousins. Polly's grandchildren and Lilly's grandchildren would be third cousins, as they share a set of great-great-grandparents but have three generations between them and a common ancestor. This continues down the line until there is no longer a common ancestor between two people.