A Relative Key means that the keys are related to one another, yet different. Like a relative in your family, same things, but different.
You may have noticed in the discussion of the Major and Minor scales that some of the Keys have the same notes, sharps and flats. For instance the Key of C Major and the Key of A Minor both have all natural notes, no sharps, not flats. The Key of G Major has F# and the rest of the notes are natural, and the Key of E Minor also has F# and all the rest of the notes are natural. So in this case C Major and A Minor are relative, as are G Major and E Minor.
So you see the similarity in they have the same notes, but what is the difference? If you read the section on Strong Chords you know that the basic structure is the 1 4 5 chord progression. This mean that the 1 Chord in C Major is the chord C Major, while the 1 Chord in A Minor is the Chord A Minor. They have different sounds and yet are only one note different. In C Major the progressions (1) C Major (4) F Major (5) G Major. In A Minor the progression is (1) A Minor (4) D Minor (5) E Minor. This difference in chords is one difference.
The other difference is the order of the notes in the scale you play. The Key of C Major starts on C and goes C D E F G A B C. A Minor starts on A and goes A B C D E F G A. Because the whole steps and half steps are placed differently the sound of the scale is different, even though they are the same notes. Same thing, sounds different, like a relative.
On nice thing about a relative scale is that you can substitute chords from each one into the other. For instance a common chord progression is G Major, E Minor, C Major, D Major. The G C and D Major are the 1 4 5 of the G Major scale, and the E Minor is the 6th of the scale, but also the 1 of the Relative Minor. This combination ends the angular feeling of the 1 4 5 and gives a more round feeling.
The other thing you can do is jump between the two relative scales with ease as they have the same notes. Major has a happy bright sound, Minor has a darker more somber sound. If you are writing a song that has both parts, you use the Major for the Happy and the Minor for the Somber. In the key of C you would play the C F G - 1 4 5 - progression until you reach the bummer part of the song, then shift to A Minor D Minor E Minot - 1 4 5 - of the relative A Minor, then shift back to the happy if you want. Or, you could take the six chords of both and mix them up to make a long extended chord run, using the notes of the similar scales to tie them together.