Some songs use minor chords extensively, so using Roman numerals from a major key doesn't make much sense. A minor key changes the way we write down chords to better suit these progressions.
A minor scale comes directly from a major key. Let's use the key of F, which is F G A Bb C D E F. If you start on the sixth note instead of the first, you get a minor scale! None of the notes will change, just the order. So a D minor scale (AKA the key of D minor) is D E F G A Bb C D. We're using the exact same notes as the key of F, but starting from a different note!
The key of D minor is the "relative" key of F major, since they are exactly the same. To figure out the relative minor key of a major scale, just go to the sixth note in the scale. Here's some others:
The key of C major is relative to the key of A minor.
The key of E major is relative to the key of C# minor.
The key of A major is relative to the key of F# minor.
Back in the lesson on key signatures, I posted a rather confusing chart of the circle of fifths. Here it is again, but notice that every major key (on the outside) has its relative minor key on the inside!
This change in emphasis to the sixth note in the scale will change our Roman numerals too. None of the chords actually change because we're still using exactly the same notes as the major scale. But they are written differently to show that we're in a minor key. Here's the key of A minor:
A C E - Minor - i
B D F - Diminished - ii°
C E G - Major - III
D F A - Minor - iv
E G B - Minor - v
F A C - Major - VI
G B D - Major - VII
So when you write a progression like i iv v in this key, it'll show that our "home" chord is really A minor.
Try writing some progressions in any minor key!