As explained in the Sidereal Time article, the Right Ascension of an object indicates the Sidereal Time at which it will transit across your Local Meridian. An object's Hour Angle is defined as the difference between the current Local Sidereal Time and the Right Ascension of the object:
HAobj = LST - RAobj
Thus, the object's Hour Angle indicates how much Sidereal Time has passed since the object was on the Local Meridian. It is also the angular distance between the object and the meridian, measured in hours (1 hour = 15 degrees). For example, if an object has an hour angle of 2.5 hours, it transited across the Local Meridian 2.5 hours ago, and is currently 37.5 degrees West of the Meridian. Negative Hour Angles indicate the time until the next transit across the Local Meridian. Of course, an Hour Angle of zero means the object is currently on the Local Meridian.