A Hero Card is a bit of a blurrily-defined category. While thematically it is obvious (it's a hero), in gameplay it can manifest its heroism in a number of different ways.
One! A Hero Card may just be a card that saves the day, regardless of whether it depicts a strapping young lad or a gigantic cataclysm. Cards that divert things like attacks, negative cards, or finishing plays often are this. This is best showed by example, so check out the image gallery below.
Two! A Hero Card may show or represent a character, that in some fashion represents the player himself. Games that use this mechanic tend to revolve around boosting the one character, using their special abilities as the main factor in strategy, and protecting them at all costs.
Three! A Hero Card may just, well, depict a hero.