If you look at some simple progressions in the key of C major, it is easy to see how the system of roman numerals works. The following example contains the chords C, F and G. Since these chords correspond to the first, fourth and fifth degrees of the C major scale, the progression could be described as I IVVI in the key of C.
This one contains the chords C, Em, Dm and G which correspond to the first, third, second and fifth degrees of the C major scale. Therefore the progression could be described as I III II V in the key of C.