The word "scale" comes from the Latin word "la scala" which means "the ladder". So a scale is a ladder of notes that starts at the bottom -- called the root note -- and proceeds upwards to the top of the ladder -- called the octave note.

There are several kinds of scales, the most common being the major scale, followed by three different types of minor scales. After that there are several specialty scales, including the blues scale used widely in jazz, R&B, blues, and quite a bit in pop music.

The "blues scale" is really a combination of the major diatonic scale (the "regular" scale we all grew up with) plus three additional notes:

The flatted 3rd;
The flatted 5th (or sharp 4th -- same thing);
The flatted 7th

As a result the blues scale really contains 11 notes -- the 8 of the normal diatonic scale -- and the 3 "blue notes". These are used in various combinations, as we shall see, to create a "bluesy sound".

The blues started not as a piano style, but as a vocal style, and of course the human voice can sing "in the cracks" between the notes on the keyboard. So when we play blues on the keyboard, we try to imitate the human voice by playing BOTH the 3rd and the flat 3rd -- BOTH the 5th and the flat 5th -- BOTH the 7th and the flat 7th. We would play in the cracks if we could, but we can't, so we do the best we can by combining the intervals to imitate the quarter steps that a human voice can sing. (Certain instruments can do that too -- for example, the trombone. Since it has a slide, it can hit an infinite number of tones between any two keyboard notes.)

So in the key of C, for example, the blues scale would include:
C, D, Eb, E, F, Gb, G, A, Bb, B, and the octave C.

In the key of F the blues scale would include:
F, G, Ab, A, Bb, Cb, C, D, Eb, E, and the octave F.

In the key of G the blues scale would include:
G, A, Bb, B, C, Db, D, E, F, F#, and the octave G.

So in improvising you can craft a melody out of any or all of these notes. Start by creating a motif out of just 3 or 4 notes, then repeat that motif as you change chords.

For example, if you were in the Key of C, you might create a motif such as C, C, G, Bb C and repeat it in various rhythms as you play the C7 chord in your left hand, then again as you move to the F7 chord, and so on.

With practice and experimentation you can play your own variety of the blues as you master the blues scale.