If you go to the New England Historic Genealogical Society website, you will find a prominent link to the Mayflower database.
So, how do you find out if you have ancestors who were Mayflower passengers? The real answer is by doing exactly what genealogists have been doing for quite some time: you do the research necessary to connect to one of the descendants of the original passengers. The advantage of having the database online is that once you get back into the mid-1800s with your research, you could check the database to see if any of your ancestors show up as descendants. If they do, then you could apply for membership in the General Society of Mayflower Descendants, commonly called the Mayflower Society.
If one of your relatives has already become a member of the Mayflower Society, you may be able to find a more recent connection and speed up the process of applying for membership. An explanation of the process is found on the Society's website on a page entitled, "Join GSMD." There are annual fees for membership in both the New England Historic Genealogical Society and the General Society of Mayflower Descendants.
Because of my own New England ancestors and other ancestors such as Mormon Pioneers, I am aware of several lineage societies I could join. Specifically, I have at least three documented original Mayflower passengers. My Fourth Great-Grandmother, Thankful Tefft, is already a documented descendant.
The significance of the database to me personally is that it provides documentation for probably thousands of people in my ancestral family. Now that it is searchable, assuming I pay the subscription price to NEGHS, I can search for specific ancestors and save a huge amount of time documenting known lines. I will likely find that I am also related to additional Mayflower passengers.