"Eligible" usually means that the people listed have met the minimum ranking criteria, which can vary from the different types of rankings (national vs. sectional, NTRP vs. age group or opens, etc).
For instance, to have a national adult age group ranking, you must play at least one of the Category I national championships to be eligible. However, your name might show up as "ineligible" in the rolling standings lists if you've played in a Category II championship and have accumulated some points.
In Sectional rankings, eligibility on a ranking list might be determined by your place of residence (do you live in the Section or just played a tournament there?), the number of points you've accumulated, or by meeting a certain number of minimum tournaments played. This can vary by each Section's rules.