The Baratheon sigil is a stag. This family contains the King of the iron throne, Robert. The stag is by the far the most hunted animal in the forest. Throughout this series, even after Robert's death, the iron throne is hunted by various others. The desire to rule even creates a war.
The Stark sigil is a dire wolf. Wolves are known to be social animals, traveling in packs. The Stark family is like the wolf in this way, constantly putting their family before others first in an effort to survive. Throughout the novel and show, the family becomes separated and when they do, each member can barely survive. It is only in the beginning of the novel, when they are all together, that the Stark family thrives. Another way in which the Stark family is connected to the dire wolf is that the wolf is practically extinct in the land of Westeros. Throughout the series, members of the Stark family get killed, becoming rare around Westeros themselves.