There are a wide variety of these organizations. There are military societies, royal societies, religious societies, town and colonial founders societies, pioneer societies and others. Some are restricted to male descendants, others to female. There are also children's societies. The vast majority require a direct line to the qualifying ancestor and plenty of documentation proving that line.
Beginning with yourself, you need to document the birth and marriage, and generally death, of each generation of the family back to the ancestor that qualifies you for membership.
To the best of my knowledge I qualify for a handful of state and national societies: The Society of Indiana Pioneers [through about a half dozen different ancestors], Son of the American Revolution [several ancestors], First Families of Ohio [member, two qualifying lines], Mayflower Society [Wm. Brewster], Sons of the Union Veterans of the Civil War, Society of the Descendants of the Founders of Hartford [CT], Flagon and Trenchor [tavernkeepers prior to 4 July 1776] and a few others.
I am currently a member of three lineage societies: First Families of Ohio [Faucett and Mahurin/Hurin families. The Roger Williams Society [descent from Roger Williams] and the Hereditary order of the Signers of the Bush Declaration.
The latter membership is through a collateral line. Some of the lineage societies offer collateral membership. It is generally the case that if your ancestor and the qualifying member of the society share the same parents or grandparents, then you qualify for membership. Example from the HOSBD below:
Hereditary Order of the Signers of the Bush Declaration: On March 22, 1775, a committee of thirty-four Harford, Maryland citizens met at the Bush Tavern and, after deliberation, signed the famous Bush Declaration. The document is characterized as the first Declaration of Independence ever adopted by an organized body of men duly elected by the people.
An applicant must furnish proof of his or her blood relationship within the second degree of kinship to a Signer of the Bush Declaration on March 22, 1775. By way of explanation see the following:
1) The applicant is lineally (directly) descended from the Signer.
2) Collateral descent (or 1st degree of kinship) through which the applicant's ancestor and the Signer have the same parents.
3) Consanguine relation (or 2nd degree of kinship) through which the applicant's ancestor and the Signer have same grandparents.
(I qualified under #2, my ancestor, Cornelius Prall Sr. and signer Edward Prall were the sons of Aaron and Mary [Whittaker] Prall.)
One advantage of applying to a lineage society is that the organization requires documented proof [and copies of the documents] of ancestry. You will know for sure that you are a direct descendant when you get done!
When applying for First Families of Ohio, I had my grandparents marriage application and submitted it as proof of marriage. It was rejected. I needed the marriage license or certificate. A trip to the City County Building Archives resolved the issue and the application was approved.
Even if you aren't interested in joining a lineage society, you might try filling out an application just to satisfy yourself that you have the line proved.
Here is a great link to a list of lineage societies. I found a few more that I qualify for!