Disclaimer: I present concepts in an order that’s a bit different from most tonal harmony curricula; I’m trying to give a progression from easiest to most advanced, without all of the other corresponding factors that go into a curriculum design.
This is not an exhaustive list, but I'm happy to make any additions suggested in the comments!
Mode mixture occurs when we mix (or "borrow") pitches/chords from the parallel mode into the current mode. This is most easily seen in what we call the "Picardy third," where an ending chord in a minor work is actually a major triad. In the below example, in C minor, we "borrow" the E♮ from the parallel major. We can also conceptualize this as borrowing the
I chord, inserting it where we might expect a
i chord. (Note: The leading-tone B♮ is standard chromaticism in a minor key, so I'm not including a separate entry on it here.)
In the following example, in major, I show a clear
I--IV--V--I progression in the opening two measures. Note how, after the double bar, I use mode mixture to use a
iv chord instead of a
Important: When using mode mixture, the mixture is kept until you reach the point of cadence. If one wanted to add a
ii7 chord after the
iv in the above example, it should really be a
iiø7 to keep the A♭ until the cadence point.
Lastly, the below example is modified from the end of Wagner's Ring cycle; it's an example of mode mixture that was later used as the 20th-Century Fox fanfare.
To create your own mode mixture, just modify a given chord (or chords) to be the quality they are in the parallel mode. (Another way to think of this: just "borrow" the chord/pitches from the parallel mode.) If you're in major and on a
vi chord, just use instead a
VI chord, since that's the harmony that's in the minor mode.
This is also basically mode mixture, but it's enough of a "thing" that I made it a separate entry. Basically, the Neapolitan is a chromatic variant of the
ii chord that's actually built on the lowered second scale degree. It is most often found in first inversion. It typically occurs in minor, but it can occur in minor or major.
Applied Chords (or "secondary dominants")
Secondary dominants occur when you precede a diatonic triad with its own
V7 chord. In the first two measures of the below example, I have a simple
I--V--I progression. After the double bar, I insert the dominant of the
V chord, marked with an asterisk. This D major chord, which is nondiatonic in the key of C major, is labeled
V/V and read as the "V of V."
Another type of applied chord is the secondary leading-tone chord. Here, I insert not a
V/V, but a
vii°7/V. The process is the same: find the chord I'm moving to, and precede it with a
vii°7 built on the leading tone of the root of that chord.
Putting It All Together
Even with just mode mixture, the Neapolitan, and applied chords, we can do some pretty gnarly stuff (listen here!):
I start with a plain old tonic moving to a dominant seventh in first inversion. With the B♭ in the bass I insert an applied chord (in inversion) to the
IV that occurs in the second measure. Then I embellish that
IV through mode mixture with a
iv (both in first inversion).
I reach the dominant, at which point I don't resolve it to tonic, but rather to an A♭7 chord in third inversion (whaaa?!?). This chord is actually an applied dominant to the Neapolitan, which appears on beat 3 of measure 3. Then I move through an applied chord to
V at the end of the third measure. I just have a big old dominant in m. 4, and then a cadence in tonic in the last measure.
There are other things I could add in here; augmented sixth chords and tritone substitutions, for instance, but I'll wait and see if the interest is there before I keep going. For now, you might just want to check out my answer at "Punning chord" to get an idea of one clever thing you can do.