Do-follow links are plain, old, ordinary links that do not contain the nofollow tag. It can be a text link or an image link.
Google is very sensitive about the use of do-follow links since they significantly impact the rankings.
As for atlrus & martinibuster, I think you both are basically saying the same thing: Google is making noise about advertorials because many of them contain unlabeled do-follow links that are slipping through their current algorithm filters. I personally can't remember seeing an online advertorial that didn't have a do-follow link.
Google tends to make noise for two reasons - #1 To warn innocent/naive webmasters so they have time to clean up and avoid the inevitable penalties & #2 To scare other webmasters with a bit of FUD (fear, uncertainty and doubt) hoping it slows down the abuse while Google develops an automated way to take care of it.
The question is how does Google identify advertorials with do-follow links (especially if the advertorial doesn't contain tell tale keywords like "sponsored post")