If you were asked to provide the relative minor key for F major, all you would have to do is count three half steps down from F to arrive at d minor. Piece of cake. Parallel keys are keys with the same tonic. For example, C major and c minor are parallel keys. When arranged consecutively in alphabetical order, the pattern of half and whole steps typically identifies a particular key tonal system based on the major and minor scales.
Scales tend to gravitate toward a single pitch, indicating the tonic pitch.
A key signature is a means of providing the correct accidentals for the scale being used, positioning the half and whole steps in the appropriate arrangement.