Extensions, and the thirds that keep going above the seventh
Ninth, eleventh and thirteenth are the stack of thirds continued past the seventh; they are colour, not structure, and the practical rule is knowing what to drop when you run out of hand.
A triad is two thirds stacked. A seventh chord, three. There is no reason to stop: stack one more and you reach the ninth, one more and the eleventh, one more and the thirteenth, which is the last note of the scale not already in the chord. Those notes above the seventh are the extensions, and popular music charts are full of them.
Why the numbers pass seven
The ninth is the same note as the second, an octave up: D, in C. The eleventh is the fourth, F; the thirteenth is the sixth, A. The large numbers say the note is above the seventh, at the top of the stack, and not in the middle of the triad.
C ninth: C, E, G, B flat, D. A dominant seventh chord with the ninth on top. The chart writes C9, and the nine alone already implies the minor seventh below it, because the numbering is cumulative: a ninth chord has a seventh, a thirteenth chord has a seventh and a ninth.
C major ninth has the major seventh; C thirteenth has the minor seventh, the ninth and the thirteenth. In both the fifth, and in the second the eleventh, are left out in practice, and that is what the next section is about.
An extension is colour, not structure
What makes a chord what it is are the third and the seventh: the third says major or minor, the seventh says dominant or not. Extensions change none of that. C7 and C13 are the same dominant chord, with identical function, and the thirteenth is a colour on top.
The second version is the first with one more note in each chord. The sentence is the same; the timbre is denser. That is why you learn the frame before the colours: a chart with many numbers is a simple chart with decoration.
What to drop
A complete thirteenth chord has seven notes, and the hand has five fingers. The practical rule is to cut from the bottom up in order of importance: first the fifth, which says nothing; then the root, if there is a bass playing; never the third or the seventh, which are the chord.
C thirteenth without the root and without the fifth: E, B flat, D, A. Four notes, and the whole chord is there — third, seventh, ninth, thirteenth. With a bass playing C underneath, nobody misses anything.
The eleventh, which collides
Over a major chord, the perfect eleventh has a problem: it sits a semitone above the major third, F against E in C, and together the two sound like a mistake. So the eleventh appears in two ways: in minor chords, where the third is minor and there is no collision, or altered in major chords, a semitone higher, F sharp in C.
The raised eleventh over a major chord is the Lydian sound, open and bright, and the sharp is not a whim: it is what avoids the collision. Over a dominant chord, the same alteration gives the sound of tritone substitution.
What to take from here
Extensions are the thirds stacked above the seventh: ninth, eleventh, thirteenth, which are the second, the fourth and the sixth an octave up. They add colour without changing function, and the numbering implies the notes below. When you run out of hand, drop the fifth and then the root, never the third or the seventh. The perfect eleventh collides with the major third, and so it lives in minor chords or turns up raised.