Answer: If you play the notes of a major scale but start on the 6th tone instead of the first, you'll hear the pattern of steps that defines the "natural minor scale." For example, the notes of the D major scale are D,E,F#,G,A,B,C#,D. Begin on B and play those same notes, and what you hear is B natural minor. So we say that B is the "relative minor" of D; it uses the same notes as D major. It will therefore have the same key signature as D major.

But if I want to make a minor scale that starts on the same pitch as a major scale, I'd have to use different notes to get that pattern of steps. A natural minor scale beginning on D would be D,E,F,G,A,Bb,C,D. That one would be called the "parallel minor" of D major: a minor scale that start on the same tonic as the major key. The parallel minor will always have a different key signature.

(Remember that in practice we usually raise the 7th degree of a minor scale and sometimes the 6th degree, too, forming the harmonic minor scale and the melodic minor scale. Those alterations are not part of the key signature. The key signature tells you the pitches of the natural minor and its corresponding major key).

To sum up: you find the relative minor of a major scale by counting up 6 scale steps (or, more easily, down 3) and playing the same pitches starting from there. To form the parallel minor you start on the same tonic but play different pitches. Major keys and their relative minor keys share the same key signature. Major keys and their parallel minor keys have different key signatures.