The blues scale, and the note that is only for passing through
The minor pentatonic with one note added, the diminished fifth between the fourth and fifth degrees; it sounds good while it moves and harsh if you stop on it, and that is what makes the blues sound like the blues.
The minor pentatonic has five notes and none of them is wrong. The blues scale adds a sixth, chosen to be wrong on purpose: the diminished fifth, between the fourth and fifth degrees. It is the "blue note", and it is the difference between a phrase that sounds correct and one that sounds like the blues.
The extra note
In A, the minor pentatonic has A, C, D, E and G. The blues scale adds the D sharp, or E flat, depending on where it is going: the black key between D and E.
That note is a tritone from the tonic. The pentatonic had removed exactly the semitones and the tritone from the seven-note scale; the blues scale gives back one semitone, between D and D sharp, and one tritone, between A and D sharp, in a single place.
It is for passing through
The rule of use fits in a sentence: the blue note sounds good while it moves, and harsh if you stop on it. It lives between the fourth and fifth degrees, and it works in both directions: going up, D, D sharp, E; coming down, E, E flat, D.
In both phrases it falls on a weak beat and resolves by semitone onto the next note. Play the same phrase stopping on the D sharp and hear what happens: the note that gave the flavour becomes a wrong note. The spelling follows the direction, sharp going up and flat coming down, as with any enharmonic note.
Over major chords
The blues scale is minor, and the blues is usually played over dominant chords, which have a major third. A blues over A7 puts the scale's C natural against the chord's C sharp, and that friction — the minor third over the major third — is the other ingredient of the sound. It is not a mistake to correct: it is the colour of the genre, and the "blue note" of the third is as characteristic as the one on the fifth.
Over that progression, the A blues scale serves throughout, from beginning to end, with no change of scale from chord to chord. That is why it is the scale of a first blues solo: one scale, for all three chords.
What it does not do
The blues scale has no leading tone and no second degree, so it does not conclude the way a major scale concludes. A blues phrase ends on the tonic because the chord ends, not because the scale pulled there. And it is not the blues' only scale: over each dominant chord, that chord's Mixolydian works too, and mixing the two — the minor blues with the chords' major thirds — is what good soloists do without thinking.
What to take from here
The blues scale is the minor pentatonic with the diminished fifth added, a passing note between the fourth and fifth degrees that works going up and coming down, and that is not for stopping on. Over dominant chords, the scale's minor third against the chord's major third is the genre's other colour, and a single scale serves all three chords of a blues.