Being stealthy in Skyrim is very important, as it allows you to pick your fights, perform a bit of skulduggery, and perhaps start out fights with a devastating sneak attack. When you enter sneak mode your status is denoted by the eye in the center of the screen that replaces the crosshair. When the eye is fully closed (looks like a line broken by a dot) you’re completely undetected. The "HIDDEN" text under the icon sometimes appears to remind you of this face.
As others (hostile or not) become aware of you, the eye will begin to open. In combat situations basically any hostile action will arouse suspicion (casting spells, shooting arrows), usually provoking the enemy to investigate, When the eye is fully open (often accompanied by the word "DETECTED") you’re no longer hidden and may suffer the consequences. Successfully sneaking is dependant upon many factors, including an enemy’s line of sight, proximity, darkness, movement speed and the weight of worn armor (and hence the sound made by moving) all play a role.