'Diatonic' essentially means 'belonging to the key / scale of'.
A key / scale is built using very specific notes - these are known as the diatonic notes of the scale, ie the diatonic notes of the scale of C Major are C, D, E, F, G, A and B - change one and it would no longer be the scale of C Major.
Diatonic chords are chords built using only the diatonic notes of a certain key, that is to say diatonic chords in the key of C Major will also only use the notes C, D, E, F, G, A and B. We build the diatonic triads (three-note-chords) by beginning on any one of the diatonic notes, then adding the note an interval of a third above that, and then another note a third above that - you could also think of this as 'play a note, miss a note, play a note, miss a note, play a note' if it is useful.
Each note in the scale can be assigned a number, and we call these numbers 'scale degrees'. ie again in C Major, C = 1st scale degree, D = 2nd scale degree, E = 3rd scale degree, and so on.
Further, if we build a chord beginning on the 1st scale degree, we would get the notes C, E, G - the notes of C Major. Therefore we can say the chord built from the 1st scale degree in the key of C Major is the chord C Major.
Building from the 2nd scale degree, we get the notes D, F, A, the three notes of the triad D Minor. Therefore we can say that the chord built on the 2nd scale degree in the key of C Major is D Minor.
We can continue this with all the diatonic notes until we have built all of the diatonic chords. There are 7 individual diatonic chords for each major scale, and 7 for each minor scale, and these are shown in the tables here.
Crucially, we can give each of these chords a roman numeral where the number of the numeral is the same as the scale degree the chord is built from, and the case of the numeral dictates the chord quality, where upper case means Major and lower case means Minor.
In major keys, such as the key of C Major, the diatonic chords always follow the same pattern of chord qualities, and we get:
I, ii, iii, IV, V, vi and vii֯ (or viidim)
This shows us that the 1st, fourth and fifth diatonic chords are Major, the 2nd 3rd and 6th are minor, and the 7th is diminished. This is the same for ALL major scales, and so we can conveniently build these into a graph.
The pattern is different in the natural minor scale:
i, ii֯ (or iidim) III, iv, v, VI, VII
We will add more to this later.