The process of mode mixture involves borrowing of chords from the parallel minor key in the major mode. These borrowed chords are essentially those which contain the scale degrees that are different between two parallel major and minor keys, namely, the b3, b6, and b7 degrees. When combining this scale degrees with some of the native chords of the major scale we get the resulting chords of bVI, iiº, iv and viiº7.
The bVI chord is a major triad on the b6 scale degree. For example: in the key of C major, this chord would be A flat major. Its main usage is as a replacement chord for the major mode native vi chord, especially on deceptive cadences. It can also be used as a subdominant chord leading to degree V.
The iv chord is a minor triad over scale degree 4, changing the normatively major IV degree into its minor counterpart. For example: in the key of D major, this chord would be G minor. It is essentially used exactly in the same ways as a normal IV chord. As a subdominant chord in cadences, in the second inversion as a contrapuntal neighbor chord, and as a good pivot chord for modulations (this will be expanded upon later).
The iiº and ii half-diminished seventh chords are chords based on a diminished triad over scale degree 2. For example: in C major, these chords would be D diminished triad and D half-diminished seventh. The main use for these chords is essentially that of being subdominant leading to V or Vii. It is very usual to find them in the first inversion due to the fact that in this position the bass note is scale degree 4, the main subdominant note.
The Viiº7 chord is a fully diminished seventh chord over scale degree 7. For example: in the key of A major, a G#º7 chords. This structure owes its origin to mode mixture since it turns the otherwise half-diminished seventh chord over scale degree 7 into a fully diminished chord by adding the b6 scale degree. Due to its symmetric nature, this diminished seventh chord can appear in any inversion and has multiple uses. Among these are, as a contrapuntal chord joining two chords by smooth voice leading; as a dominant chord leading to I and last but not least, as an ideal chord for modulation (this idea will be expanded later).
Another common type of mixture chord is the bII, also referred to as the Neapolitan chord. This chord is essentially derived from the mixture with the Phrygian mode which has a naturally occurring scale degree b2. At its core, the Neapolitan chord is a major triad occurring on the b2 scale degree.
The Neapolitan chord is usually found in minor keys and in the first inversion. This close association with the minor mode can be attributed to its keener relationship to the sound collections found in these types of keys, essentially, the b6 scale degree which is native to the minor mode and is the fifth of the bII Neapolitan chord. The first inversion usage is better explained from a functional point of view. Fundamentally speaking, when the Neapolitan chord is in the first inversion, the bass note is scale degree 4, which is the main representative for the subdominant function. This fact leads to the strong pre-dominant function that the bII chord exhibits but with the special color that the b2 scale degree adds.
As you may have already noticed, most of these chords are members of the subdominant function. This fact is one of the main factors for the great usefulness that mixture chords display to produce modulations to foreign keys. Usually, for pivot chords, the best choices are subdominant chords since these are the least defined in terms of tonal resolution. While the dominant has an enormous amount of pulling towards the tonic, and the tonic has stability, the subdominant is less characteristic of any particular key and is, therefore, more susceptible to being reinterpreted in other keys.
When using mode mixture, the tonal spectrum broadens and the flat side of the circle of fifths becomes closer than before. For example, in the key of C major (no accidentals), the iv chord opens up possible modulatory space to go to flat keys like E flat major (3 flats), where it is ii; or A flat major (4 flats), where it is vi; or even D flat major (5 flats), where it is iii. Furthermore, by using the viiº7 chord the tonal space broadens even more since due to its symmetrical structure this chord can be used as the viiº7 chord in the keys of C and Cm, Eb and Ebm, Gb and F#m, and last but not least, A and Am. All of these keys are foreign to the original C major, a possibility that was nonexistent when using purely diatonic triads from the major scale.
To conclude, mode mixture is an enormously useful tool for composition and elaboration of simple chord progressions. They can add color to otherwise plain chords, they can be useful to modulate to very distant keys and they can even be utilized to exploit the rhetorical duality between major and minor tonality. By using these new chords, the tonal spectrum is broadened, and the possibilities are unlimited.