The Stormcloaks are part of the Civil War Quest Line. They are a group rallying around Ulfric Stormcloak, the High King of Skyrim, in an effort to make Skyrim independent from the Empire seated in Cyrodil. He claimed the throne by killing the previous High King in a duel in the old ways. Some claim that Ulfric's use of the Thu'um to knock the High King Torygg on the ground before stabbing him with a sword to finish him was not consistent with traditions while Ulfric contends that the Thu'um is merely a skill that anybody can learn and that the killing blow was made with a sword while the Thu'um was merely an aid.
Some consider Ulfric's claim to the throne is illegitimate, especially when the previous High King was open minded to Ulfric's wishes and even respected him. Other's suggest that Ulfric's challenge of a teenage ruler to duel when he was in his prime and had spent years studying the thu'um was savage and murder. However, many Nord's, regardless of their stance on Ulfric's position as High King are sympathetic to his cause. Amongst his men, known as the Stormcloaks after their admiration for Ulfric, he is known to be a charismatic and fearless leader who is so powerful that he "shout[ed] the previous High King to pieces".
For more on the series of quests, please see the walkthrough.