The short explaination is that the church modes are the notes of the major scale starting on different scale degees So if we are talking about the church modes based on teh key of C the modes would be ionian C - C dorian D - D phrygian E - E lydian F - F mixolydian G - G aeolian A - A locrian B -B As far as jazz goes it is really just a convenient way to talk about a specific group of notes and you can make alterations to the modes for example G mixolydian (#11) would be G A B C# D E F G This is not a church mode but the terminology makes it easier to describe that set of pitches.