The numbers stand for the chords of a certain key. For instance, the I chord in the key of C is C major. The IV chord is F maj, and the V chord is G maj. In the major scale the chords go like this: maj, min, min, maj, maj, diminished. So in C you have:
I - C maj
II - D min
III - E min
IV - F maj
V - Gmaj
VI - A min
VII - B Dim
The reason for this is that each chord contains three basic notes, the triad of the root (first note), the third (determines whether it is major or minor) and a fifth. To stay in the key of C, for each chord, this triad has to be built from notes within the key. So if you play a G chord you play G (the root), B (the major third, considered a mjor third if it is to whole tones up froom the root), and D (the fifth).
The chord of A has an A (the root), C (the minor third, which is called a minor third because it is only a tone and a half up from the root), and E (the fifth).
The chords are major or minor to keep the third note within the key. A diminished chord has both a lowered third, and a lowered fifth, in the case of B, an F, to keep it within the key of C.
Hope this helps, it might be a bit in depth, but any questions feel free to ask.