The best way to understand the modes is to hear them side by side, starting from the same root.
Normally, when an explanation of the modes is given it goes something like this (I start by showing you this, which is the *wrong* way to explain modes)
C Ionian is C D E F G A B C
D Dorian is D E F G A B C D
E Phygian is E F G A B C D E
....and then the student plays a C Ionian, starting on C and ascending and descending the scale, and then the teacher (or website, or whatever) says "ok - now to the Dorian mode!" and the student then plays D Dorian, starting on D and ascending, descending, etc... all the examples are in the same key - C Major, just starting on different notes, so naturally, the beginning player scratches his or her head and goes "huh? it's just starting on different notes on the scale" and they don't *hear* the benefit of the modes.
The BEST WAY TO UNDERSTAND THE MODES, as I stated earlier, is to play them all starting from the same root - in other words, all starting on C, for example (consult any good website near you for fingerings)
C Ionian C D E F G A B
C Dorian C D Eb F G A Bb
C Phrygian C Db Eb F G Ab Bb
C Lydian C D E F# G A B
C Mixolydian C D E F G A Bb
C Aeolian C D Eb F G Ab Bb
C Locrian C Db Eb F Gb Ab Bb
By doing this, you will be able to hear the difference in the sound of the modes much more clearly.
In terms of which modes to play over which chords, I'm not super expert on this, but generally, you match the interval between the 1st and 3rd of each scale with the Major/Minorness of the underlying chord.
Example - chord is C major, I can play C Ionian, C Lydian, or C Mixolydian over it because the interval between the 1st and 3rd of each of these scales is a MAJOR 3rd.
Example - chord is C Minor, I can play C Aeolian, C Dorian, C Phrygian or C Locrian over it, because the interval between the 1st and 3rd of each of these scales is a MINOR 3rd.
Example - chorsd is C Diminshed - I can play C Locrian over it because it contains the MINOR 3rd and a DIMINISHED 7th interval from 1st to 3rd and 1st to 7th respectively.