Pentatonics, and the scale where nothing sounds wrong
Five notes instead of seven, chosen so that no combination clashes; the scale of almost everyone's first solo, and the reason it is so safe and so directionless.
Take the major scale and remove two notes: the ones that form a semitone with a neighbour. Five are left, and those five have a property no seven-note scale has: any pair of them sounds good together. This is the pentatonic, and it is the scale almost everyone plays their first solo on, because there is no note in it that clashes.
What was removed, and why
In the C major scale, the semitones fall between E and F, and between B and C. F and B are the two notes that form those semitones with a neighbour, and they are also the two that form the tritone with each other.
Remove them both and what is left — C, D, E, G, A — has no semitone and no tritone. The two intervals that create most of the tension in the major scale left together, and with them went the possibility of a wrong note: any note of the pentatonic over any chord in the field of C major sounds, at worst, acceptable.
The minor pentatonic
The same five notes, starting on A, are the A minor pentatonic: A, C, D, E, G.
It is the relative relationship applied to five notes: the C major pentatonic and the A minor pentatonic are the same set with a different centre, exactly as C major and A minor are. Compared with the natural minor scale, the minor pentatonic is missing the second and sixth degrees — the ones that formed the semitones.
It is the scale of the blues, of rock, and of a good part of popular soloing, and the reason is the same: over a minor chord, or over a blues progression, none of the five notes falls outside.
The price: nothing pulls
Safety has a cost. The semitones that were removed were exactly the notes that gave the scale its direction: the leading tone, which pulls to the tonic, and the fourth degree, which pulls to the third. Without them, the pentatonic asks for resolution nowhere. You can stop on any note, and none of them sounds like an obligatory ending.
Sing both. The second one ends because the B asked for the C; the first one merely stops. That is why a solo made only of pentatonic can start to sound unmoored after a while, and why improvisers add the missing notes — the leading tone and the fourth degree — when they want a phrase to arrive somewhere.
The blues note
The blues adds a sixth note to the minor pentatonic: the raised fourth degree, D♯ in A, between the fourth and fifth degrees. It is the "blue note", and it gives the scale back exactly what the pentatonic had removed — a semitone and a tritone — in a single place, to be used in passing.
What to take from here
The major pentatonic is the major scale without the two degrees that form semitones; the minor pentatonic is the same notes starting from the relative minor. With no semitones and no tritone, no note clashes, which is why it is the scale of the first solo. With no semitones there is also no direction, which is why it is a starting point and not a destination. The blues scale gives one semitone back, deliberately, in the right place.