This seems quite related to your earlier question, and my answer to it.
In that earlier answer I illustrated why, if you take a "general" direction of rotation for an arbitrary object, the angular momentum will not be aligned with the angular velocity. However, for any object there are some special directions for which they will be aligned. These directions are often axes of symmetry of the object (although they don't have to be in the case of objects with no obvious symmetry). In the case I illustrated in my earlier answer, the long axis of the rod would be a "principal axis". Such an axis has the nice property that it's easy to calculate the angular momentum when you know the angular velocity.
In the context of your question, $I$ is once again the inertia tensor: that is, a tensor that allows you to compute, in general terms, the angular momentum of an object given the angular velocity. An inertia tensor can be decomposed into a rotation matrix and a diagonal matrix, the elements of which correspond to the principal moments of inertia. This is explained in the subtopic principal axes on the Wiki page for "moment of inertia".