Yes, this post breaks the century mark. But on to more important things: common-chord modulations. The concept of modulation is easy enough. At some point the tonal center has shifted, so a new note is the goal of the melody and harmony. The idea of common-chord modulation (or pivot-chord modulation) is a little more complex, but the general definition is still simple. The shift of the tonal center is created by using a chord that is found in both keys. An A minor triad is found in both C major and G major. The difference is in the expectations for how that chord will resolve. In a pivot-chord modulation, the A minor triad resolves to a chord that fits in the key of G major, not in the key of C major. So if we start a piece in C major, bring the harmonic progression to an A minor chord (vi for those playing at home), and resolve that chord to a D major chord, we no longer feel like we are in C. Instead, the motion from the A minor triad to the D major triad evokes the sensation of ii - V, leading us to the expectation of a tonic chord in G major. The D major chord doesn't fit in the tonality of C major, signalling that we have shifted keys. The A minor chord is the common chord, pivoting us from the key of C major to the key of G major.
The problem comes with deciding exactly which chord is the pivot. Typical modulations involve closely related keys that share many chords. The musical context can create very ambiguous situations where there seem to be many possible answers. There is usually one best answer, but it is often difficult to communicate to students why that answer is best. There are a plethora of variables involved, most often resolved by "the ear." The musical ear or musical intuition is really the application of an ingrained sense of the rules governing music. Delineating these rules can be more confusing than enlightening. That is why theory courses are always accompanied by aural skills courses. These classes on "ear training" try to engrain the rules of tonal music in the students, without listing all of the rules. Singing countless melodies, transcribing melodies and harmonies, playing scales and progressions at the piano: these are the activities that are used to create a fluency in the language of tonal music. It is really like the natural language acquisition of young children. They listen to adults, mimic adults, and eventually attempt to put words together into sentences. Hmm, maybe a look at theories of language development could inform aural skills pedagogy. I think I smell a paper topic!