Birth, marriage and church records, wills, tax records, family bibles, census records and a number of other sources such as DNA tests......can provide evidence of a person's relatives.
My father operates our "Easterling Family Genealogical Society, Inc." and has used all of the above mentioned sources to determine over 20,000 familiar relationships.
He also makes use of the extensive records maintained by The Church of Jesus Christ of Latter-day Saints (Mormons) in Utah, even though he is a Southern Baptist.
The EFGS's most recent efforts have utilized DNA test results to establish proof positive of "blood" relatives of our American family members with those who live in England.
I edited the first edition of our 1,000+ page "Easterling Family History Book" which was published in 1990 and quickly found that about half of one's relatives do not have the same "last name" because the female "blood" relative's names change when they marry.
Without blood tests? Tax records. Parents generally claim their children as dependents and have to list the social security number on the 1040.
Ask their mother.
There are a number of methods to determine if two people are related. Public records (birth certificates, marriage and church records, wills, tax records, family bibles, census records, etc.) have traditionally been used to identify a connection. More recently, DNA tests have provided a more scientific method of identifying a shared genetic background. Learn more at 23andme.com.