The Erdös number is determined by stringing together collaborators on published papers.

That is, Paul Erdös wrote a paper with Alan Hoffman, who wrote a paper with Joseph Kruskal, who wrote a paper with Mark Liberman, with whom I've collaborated on a couple of papers. My cousin Tom Sag's Erdös number is 2.

Check out the Erdos Number Project Home Page, or read Erdös's biography, The Man Who Loved Only Numbers.