Dragon Skills are a gameplay element used in Stable Missions. Each dragon has two (2) skills and no dragon has the same two (2) combinations of skills. Some dragons have unlisted skills like the skill "Leader" (which, the Night Fury seems to be the only one with this skill).
To learn more about Dragon Skills and Stable Missions, please visit the Stable Missions page.
As for 2.13, the Dragon Skills are now important for Dragon Tactics, the 1st Skill is their Role while the 2nd Skill is their Element.