I believe this is the most misunderstood concept in music theory , and I don't know a good reason for this misunderstanding since it was meant to ease learning and memorizing theory topics.
Anyway , I'll try to give you a simple explanation for this .
First you have to know that the main scale in western musical theory is the major scale , which is based on this formula [w,w,h,w,w,w,h] .
Let's take the simplest example for a major scale which is the C major scale , the notes in order are [C D E F G A B c] ... The main thing to make this scale sound like a major scale is to start and end on the C note , which is called the root note.
What if you changed that , what will happen ? It will give a new effect , just try it out .
Take the same scale with the samew exact notes , start & end on an A note instead ... [A B C D E F G a] , now that's an A minor scale (aeolian mode) , which is the 6th mode (or degree) of the major scale (key) .
That's how you derive a mode from the major scale (or key) , you simply change the root note which is the starting & ending point of the mode or scale .
So can you list the modes derived from the C maj key ?
[C D E F G A B c] C major , or Ionian mode
[D E F G A B C d] D dorian mode (from the C maj key)
[E F G A B C D e] E phrygian mode (from the C maj key)
[F G A B C D E f] F Lydian mode (from the C maj key)
[G A B C D E F g] G mixolydian mode (from the C maj key)
[A B C D E F G a] A aeolian mode (from the C maj key) , aka natural minor scale
[B C D E F G A b] B locrian mode (from the C maj key)
So , from any major scale (key) you can derive the seven modes by changing the root , and their names in order starting with the major scale are : Ionian , Dorian , Phrygian , Lydian , Mixolydian , Aeolian , Locrian .
