There are 2 common ways to construct the blues scale - either using the major or the minor pentatonic scale of a key.
The blues scale is made from the 1st, flattened 3rd, 4th, flattened 5th, 5th and flattened 7th notes from the major scale for a given key.
The music theory term flattened means to lower the note by half-tone / semitone / piano key.
The flattened 5th note of the major scale is the blue note that gives the scale its unique character.
The 7th and final note in the diagram is the octave note, named the same as the tonic note, and is the note where the scale starts repeating all the way up the piano keyboard.
For example, in the key of E-flat, below is the Eb blues scale.
To construct the blues scale from the minor pentatonic scale, take the 4th note of that scale, flatten it, and insert it before the 4th note position of the same scale. This 4th note is the blue note.
Yet another (more complex) way to identify the blue note is to take the Diminished 5th note interval based the tonic note, but it is usually easier to work it out from scales rather than intervals.
The blues scale links at the top of this page are all keys from the Circle of fifths diagram.
Some of these keys are shown on the circle of 5ths diagram as being major, minor or both.
To see the blues scale construction using a major scale, choose a link above that is shown in Red on the circle of 5ths diagram, eg. Cb blues scale
To see the blues scale construction using a minor scale, choose a link above that is shown in Green on the circle of 5ths diagram, eg. A# blues scale
Some keys are both major and minor, in which case both major and minor constructions are shown, for example - D blues scale.