The Roman numeral system is just another way of referring to the chords derived from a given scale. I will use the C major scale as an example: C, D, E, F, G, A, B, C. All seven tones in the C Major scale can be "arpeggiated" to create corresponding chords, all of which are related to the C Major Scale. Each one of those chords is given a corresponding Roman Numberal: I, ii, iii, IV, V, vi, vii. The 'I' chord always makes a Major 7th chord. The ii and iii always make a Minor 7th Chord. The IV makes a Major 7th Chord, the V makes a Dominant 7th Chord, vi makes a minor 7th chord and the vii always makes a minor 7 with a flatted 5th. Lowercase roman numerals are used for 'minor type' chords. Uppercase roman numerals are used for 'Major type' chords (Major and Dominant Chords)

Using the C Maj scale:

I = CMaj7 (C, E, G, B)
ii = Dm7 (D, F, A, C)
iii = Em7 (E, G, B, D)
IV = FMaj7 (F, A, C, E)
V = G7 (G, B, D, F)
vi = Am7 (A, C, E, G)
vii = Bm7b5 (B, D, F, A)

So, for example, when you hear people referring to the ii, V, I progression (typical jazz chord progression) They are referring to a progression which will use the ii, V, and I chord derived from a given Major Scale. Using the C Major Scale, a ii, V, I progression in the key of C would involve the Chords Dm7 (ii), G7 (V), and CMaj7 (I). If you played any notes from the C Scale over that progression, they would sound completely in tune since all of the notes in the above chords are contained within the C scale.

As for the use of sharps and flats, some notes will be listed with more than one 'name' (F#, Gb) because both types of terminology are necessary when you are forming scales and chords. You don't see both sharps and flats used when forming the basic major or minor scales because using both sharps and flats could violate the rules necessary to denote different scales and chords. For an example of what I'm talking about, check out the posting titled Bb7(13). James talks about the notes necessary to form that chord. (Bb, D, F, Ab, G). A book he read indicated that there was a G# in the chord. That is not true. Although (G#, Ab) have the same sound, they are not the same note. G# would be a note based on the 6th tone of the Bb scale. Ab is a note based on the 7th tone of the Bb scale. A Bb7 chord involves the Root, 3rd, 5th, and FLATTED 7TH, not the raised 6th. Its mostly semantics, but to refer to the notes incorrectly in a given scale or chord could make things completely confusing. Now there are more exotic scales and more complex chords which may contain both sharps and flats, but since you are working with beginner students, they don't need to concern themselves with that right now.

[Edited by trebledamage on 12-12-2001 at 05:30 PM]