That means for that contact that there is 1 or more other accounts or contacts with similar information that you can link that contact to...

Ex. I have 2 friends that are husband and wife and since their "home phone" and "address" are the same, it is giving me the option to "link" them into one contact.

If you tap the button, it will show you all potential "links" for that contact.


EDIT: Sorry, but I jumped the gun. This info still holds true in the "People" app, but for what you are asking it is incorrect!

 

I believe it is new facebook posts or other updates (like twitter) or events (like birthdays). If you tap the number or open up your contacts, scroll over to updates and events.

 

It means there are that many new notifications associated with the person. If you click on the name of that person (not their picture, their name underneath the pic) it will open that person's contact card and you can scroll left or right at the bottom to see what the notifications are. It may be a new message, update, event, photo, etc. It's letting you know there is something new with that person. Open their contact card to find out what it is.

 

Yeah , just color me mistaken!
I missed the part where the OP was talking about the widget!

Sorry (JJ crawls back into the corner)