We will try to keep it simple.
If I have a major scale, for instance, C Major, that scale has a key signature, which tells us how many flats and sharps to play. C has no sharps or flats, so it's easy to talk about, and also to see on a piano. Just imagine playing the white keys and only the white keys.
Now, to play C Major we need the note C to be the 'home note', or tonic. That's the note that feels like home when we resolve to it.
But it's possible to play all the same notes, as if we are in C, but instead of C feeling like home a different note feels like home.
So if we aren't playing any sharps or flats, but the note G feels like home, then we are playing a mode, or variation of a major scale.
In other words, we are playing the same notes but in a different order.
Imagine starting the alphabet at M, and when you get to Z you start immediately back at A, and finish up at L. It's a variation of the Alphabet - it has all the same letters, we just changed the starting point.
On a piano, it's playing all the white keys but instead of playing C to C, we would play G to G (like our example above, when G is home). So the scale becomes:
This is different than G major. G major starts and ends on G but has an F# in it. Remember, we are still playing the white keys, we just changed our starting point.
Every mode has a name.
In the example above we begin on the 5th note of the major scale, and the name for the 5th mode is Mixolydian.
The second mode would start on the second note of the scale and is called Dorian, which is often the first mode I teach my students. Hopefully now you can see that the Dorian mode of C Major would be D (the second note of the scale).
If we were working in G Major we would have a key signature of one sharp. The Dorian mode of G would start on the note A because A is the second note of G major.
Therefore, G major and A Dorian share that key signature of one sharp.
Another way to think about it is that A Dorian is a derivative of G major.
When you begin to grasp this concept of the tonic shifting to a different note even though the key signature remains the same, then you are ready to examine and understand modes by the first method: they are derived from the major scale.
However, there's another valid way to build them, if you understand and have memorized the structure of a major scale.
There are seven notes in the major scale, and therefore seven major scale modes.
A rose by any other name would smell as sweet, and a mode by any other name would sound the same. But for those of you geeks who like this stuff, here's the low down:
Mode 1: Ionian (Also known as the major scale)
Mode 2: Dorian
Mode 3: Phrygian
Mode 4: Lydian
Mode 5: Mixolydian
Mode 6: Aeolian (Also known as the natural minor scale)
Mode 7: Locrian