Here's my stab at an intro below.  I don't think the first two paragraphs would scare off the average lay person, but I think some mention of the formal definition needs to be made.  Let me know what you think. 
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 Angular momentum is a quantity that is useful in describing the rotational state of a physical system.  For a rigid body rotating around a fixed axis (e.g. the fins of a ceiling fan) the angular momentum depends how the mass is distributed in the body as well as the body's rate of rotation.  Angular momentum is sometimes described as the rotational analog of linear momentum. 

 Angular momentum is conserved in a system where there is no net external torque, and its conservation helps explain many diverse phenomena.  For example, the increase in rotational speed of a spinning figure skater as the skater's arms are contracted is a consequence of conservation of angular momentum.  The very high rotational rates of neutron stars can also be explained in terms angular momentum conservation. Moreover, angular momentum conservation has numerous applications in physics and engineering (e.g. the gyrocompass). 

 Fomally, the angular momentum of a point object is defined as the cross product of the object's position vector and linear momentum vector ().  Angular momentum is a pseudovector whose magnitude is given by  where  is the angle between the object's position vector and velocity vector.  The direction of the angular momentum can be determined by applying the right-hand rule.  The angular momentum of a system of particles (e.g. a rigid body) is the sum of angular momenta of the individual particles.  For a rigid body rotating around an axis of symmetry, the magnitude of its angular momentum can be expressed as  where   is the body's moment of inertia about the axis of rotation and  is its angular speed.