Three voices, stacked in thirds
A triad is three voices: a root, a third above it, and a fifth above that. Stacked on C, the gaps between neighbouring voices are thirds. C to E is 4 semitones, a major third; E to G is 3, a minor third; so C major is 4 + 3. The other triads are the other ways to stack those two gaps: C minor is 3 + 4 (C Eb G), C diminished 3 + 3 (C Eb Gb), and C augmented 4 + 4 (C E G#).
One voice at a time
Read from C major, each of those triads is one or two voices stepping by a semitone while the rest hold. The third down a semitone, E to Eb, gives C minor. The fifth up a semitone, G to G#, gives C augmented. The third and the fifth both down a semitone give C diminished. The root never steps: a chord keeps its name because its root holds, and the voices it shares with C major are its common tones.
Trading the third: sus chords
A suspended chord has no third at all. Csus4 puts the 4th, F, in its place, one semitone above E; Csus2 puts the 2nd, D, two semitones below it. With the third gone the chord is neither major nor minor, and the power chord, C5, goes one further and keeps only the root and the fifth.
A fourth voice: sevenths and sixths
Stack one more third on the triad and the fourth voice is a seventh. Bb, 10 semitones above C, is the minor seventh that makes C7; B, 11 semitones above, is the major seventh that makes Cmaj7. On C minor the same two give Cm7 and Cm(maj7). In Cdim7 the seventh is Bbb, 9 semitones above the root, so every gap in that chord is 3 semitones. A sixth, A, also 9 semitones above C, is added to a triad without a seventh: C6 and Cm6.
Above the octave: the 9th, 11th and 13th
Keep stacking thirds past the seventh and the next tones land above the octave: the 9th (D), the 11th (F) and the 13th (A). They are the 2nd, the 4th and the 6th an octave higher, so on the pitch-class clock, which folds every octave onto twelve positions, they sit where those notes do. A chord named for its top tone carries tones below it: C9 holds the minor seventh and the 9th, and C13 the seventh, the 9th and the 13th. An add chord adds only the tone it names: Cadd9 is C major and D, with no seventh.
Altered tones
An altered tone is one of those tones a semitone off: the flat 9th and the sharp 9th on either side of the 9th, the sharp 11th above the 11th, and the flat fifth and the sharp fifth on either side of the fifth. C7b9 is C7 with Db added; C7#11 adds F# and keeps the G beside it.
Reading the table
Degrees name each tone by its place above the root, so b3 is the minor third and b7 the minor seventh. Semitones count the same distance on the clock. The last column says which of C major's voices step or give way and what is added; every chord links to the builder, where any root can be picked.