Here's what I could find out. But, like the supernova that results when real stars die, the information that's available on the subject is kind of nebulous.
Back in 2007, the NFL proposed to have a C captain's patch to designate who the various team captains were. The NFL coaches liked the idea (even though NFL captains just call the visiting coin flip, and say whether they will kick, receive, or defer - yip skip). But, they also wanted to honor captains who had longer tenures as captain, back in the day when there weren't captains for every position (teams can have up to 6 captains).
So, they decided to allow a captain to wear the patch and a gold star for each year they have been designated a captain under the new system. They then said that a perennial captain (one who had been a captain before the new system) could wear a white star for each year they had been designated a captain in the old system, with a maximum of four stars on the patch.
And, the meaning of the stars is not a hard and fast rule handed down by the NFL. Teams can basically do what they want with the stars, and the significance might (and probably does) vary from team to team.
If that's not correct and someone can point us all to a link that gives a definitive answer, I'll stand corrected. But, after scouring through articles on ESPN and other sports sites, and a blog called uniform watch (http://www.uniwatchblog.com/ ), that's about the best description of the significance of the stars that I could come up with.