The first column shows the key or scale that the other chords are taken from. Let’s take the key of C (or C Major) as an example.
The other columns show the degrees of the indicated scale or key. A “degree” is the note that’s in a particular spot within the scale, or its order or location. There are seven notes in a scale, so there are seven degrees-- but we can keep counting up further to include the notes of the next octave, as follows:
Key of C Major:
1 = C
2 = D
3 = E
4 = F
5 = G
6 = A
7 = B
And if we continue counting into the next octave:
8 = C
9 = D
10 = E
11 = F
12 = G
13 = A
14 = B
These degree numbers are used when we talk about the intervals between two notes, with some intervals being called “perfect” because there is normally only one version of them, and other intervals normally having two possible versions-- “minor” and “major"-- as shown in the following example for the key of C Major. I think the abbreviations shown are standard; at least, they’re the abbreviations that I learned. Notice that since these are intervals, they refer to the distance between two notes-- from the note which is the “root” of the scale, to the indicated note.
Key of C Major:
C = P1 or perfect unison (that is, two identical C notes playing in unison, with 0 semitones between them)
C# or Db = m2 or minor second (1 semitone above the root note)
D = M2 or major second (2 semitones above the root)
D# or Eb = m3 or minor third (3 semitones above the root)
E = M3 or major third (4 semitones above the root)
F = P4 or perfect fourth (5 semitones above the root)
F# or Gb = TT or tritone (6 semitones or 3 tones-- "tri tone"-- above the root)
G = P5 or perfect fifth (7 semitones above the root)
G# or Ab = m6 or minor sixth (8 semitones above the root)
A = M6 or major sixth (9 semitones above the root)
A# or Bb = m7 or minor seventh (10 semitones above the root)
B = M7 or major seventh (11 semitones above the root)
C = P8 or perfect octave (12 semitones above the root)
Notice that there's a difference in capitalization between some of the interval abbreviations-- m2 versus M2, m3 versus M3, and so on-- so you need to be careful when writing or reading them, or you might get them mixed up.
There are also two more types of intervals-- "diminished" (abbreviated with a lowercase "d") and "augmented" (abbreviated with an uppercase "A"). These lower (diminish) or raise (augment) any of the "perfect" degrees by a semitone. For the degrees that already have two versions-- minor and major-- the diminished version lowers the minor interval by a semitone, whereas the augmented version raises the major interval by a semitone. So putting all of that together, we get the following:
Key of C Major:
C = P1 or d2
C# or Db = A1 or m2
D = M2 or d3
D# or Eb = A2 or m3
E = M3 or d4
F = A3 or P4
F# or Gb = TT or A4 or d5
G = P5 or d6
G# or Ab = A5 or m6
A = M6 or d7
A# or Bb = A6 or m7
B = M7 or d8
C = P8
We can also refer to a particular degree as being either "sharpened" or "flatted," such as b3 or a flatted third, or #5 or a sharpened fifth. I'm not certain how that works with the two-version degrees (2, 3, 6, and 7). In particular, I'm not certain whether those intervals vary depending on whether the key is major or minor. That is, in the key of C Major the 3rd degree has an interval of M3 from the root, so b3 would be the same as m3. But in the key of C Minor the 3rd degree has an interval of m3, so I think d3 might be the same as M2, in which case #3 would be the same as M3-- but I'm not certain of that.
Anyway, that chart you were asking about shows which chords we get if we start with a given degree of a key and play a chord made up of that degree plus two more notes, skipping a degree in between. For example, if we start with the 2nd degree of the key and add two more notes to it, skipping a degree each time, we get the 2nd, 4th, and 6th degrees. In the key of C Major those would be the D, F, and A notes, which form a D Minor chord. Starting with the 3rd degree, we would get the 3rd, 5th, and 7th degrees, or the E, G, and B notes, which form an E Minor chord. And so on with the rest of the degree columns shown in that chart.
Notice that the degrees are always based on the key we're playing in, so if we're playing in the key of C Minor then the same degrees might give us different notes and different chords.
I'll stop there, because it's already a lot to digest, but hopefully that's enough to help you puzzle out that chart.
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