The chainings or concatenations (encadenamientos) refer to the groups of modes that derive from the main (root) scale building the whole family. So in the most basic form, which is the diatonic chaining, you start from the diatonic scale (ionian mode) that has seven notes following the pattern of intervals (1 is a whole tone, ½ obviously a half tone) 1-1-½-1-1-1-½-1 (C-D-E-F-G-A-B-C) take into account that the sixth mode of that chaining is the aeolian mode which also happens to be the "natural minor" scale. This is important because when we speak of minor tonality we're not really speaking of a different thing than the diatonic chaining but starting on a different note. For C diatonic major the starting note to make it natural (or diatonic) minor would be starting on Eb. This makes that the Eb tonality is the ascending parallel of C. Both tonalities are siblings and when moving from one into the other, you're not actually modulating but "false modulating". This is one of the relationships that make tonalities not being separate entities but groups or families that spread a lot the range of chords and modes that you can use in your playing/composing/improvising. This kinship is important because it would allow you to play the same pentatonic scale on all the modes of all the tonalities within the family (politonal) but one.
Harmonic chaining derives from the idea of the harmonic minor scale which is a natural minor (aeolian) replacing the 7b for a 7. This note swap transposed to the first degree of the chaining (ionian mode) leaves us with the harmonic major scale. For C it would have the C-D-E-F-G#-A-B notes which is the interval sequence of 1-1-½-1&½-½-1-½. If you develop all the modes derived from that ionian scale you'll see that the 1&½ interval moves to a different degree of each mode, which makes that the chords (triads and 7th ones) in every mode are different from the ones in the diatonic chaining. This opens a new whole spectrum of secondary dominants, symmetric chords, inversions and tensions that you can use to produce sensations and moods, for instance that the 3rd degree (phrygian mode) instead of being a minor one as it is in diatonic, becomes a major mode, with a major chord that is dominant and that includes alterations (9b, 9# and 6b). Depending on the mode you use, harmonic chaining melodies sound flamencoish, arabic or manouche with a slight modification. That modification leads us to the next chaining family which is
Neapolitan chaining. It derives from the concept of using a "blue note" and/or a major 7th in the 3rd degree (phrygian mode) of the harmonic scale. This produces the funny effect of the chords in that chaining moving around a dominant chord instead of a ionian chord, so the rest position of the tonality shifts to a different degree without a real modulation if you play those notes as color notes of the minor chords involved.
These three chainings, diatonic, harmonic and neapolitan are all considered a part of the diatonic conception of western music and have produced most of the harmony that we can hear on classical music up to Beethoven.
The other two chainings derive from the melodic scales. The melodic major is kind of complementary of the harmonic minor, both include a 1&½ intervals at some position of their modes and offers a couple of very interesting features, they being that the phrygian mode is major (hence a dominant) but has not a 4th note so it doesn't ask for resolution, and that we have a minor subdominant (minor lydian mode). The melodic minor chaining is pretty complex because it doesn't have a ionian (rest) mode, it's mostly comprised of dominant chords having tensions in unusual degrees, very interesting ones as in semi-altered chords (dominants with a 2 and a 6b or 2b with a 6). On classical (Berklee) harmony most of its modes aren't contemplated as a part of a whole sequence of modes into a chaining but as separate scales that you could use when certain dominant chords are in the harmonic progression. So what I see as an aeolian major they name it as a 6b mixolydian. The difference is huge because knowing the chaining and the degree that the mode you're into (or leading to) is into the "whole", allows you for a quicker and easier chords substitution or location in the fretboard of the mode that is convenient to play. It also allows you to understand why you can use tritone substitution on dominant chords, and also the most interesting thing of all, implying without playing, which is suggesting with your melodies something in the harmony that isn't really sounding, and vice-versa. That's something that Stevie Wonder absolutely masters.
This conception makes an easier learning of the secondary dominants and the scales you'd be able to use when they're in the progression, also makes a lot easier modal modulation, and helps much more to understand and use voice movements since the change of a single note into a chord, takes you to a new and different set of harmonic and melodic possibilities.
One of the most important things to keep in mind is that the bass rules. You could be playing on your guitar a wonderful second inversion of FMaj7 (FMaj7/A which on guitar is also Am7/6b) that if the bassist is playing F it will pass unnoticed, so what would be in your arrangement a movement into the harmonic area of the dominant, would clearly remain in the harmonic area of the tonic.
