LARS LUNDIN and his database are matchmakers of a sort, trading in information about matches made long ago. Genealogists send him family trees, and his computer compares them with other family trees in a database of more than two million current and former Danish citizens. Each month, he estimates, his computer finds more than 1,000 connections between cousins. Projects like Mr. Lundin's are becoming increasingly common as both amateur and professional genealogists use the Web to pool resources, compare notes and connect family histories. Some, like Mr. Lundin, do it for fun, while others are building companies that offer higher-quality...