Ok, this needs a little bit of cleaning up. here goes:
There are two ways to view the modes: as being parallel or relative to one another.
When using the modes in a parallel way, you are looking at every mode as being an entirly different scale. so, if you had a C5 chord, you would play every mode over it STARTING ON C!! thus, C ionian, C dorian, C phrygian, etc.
The other way is the way discussed so far, called relative. D dorian is the scale starting on the 2nd step (or degree) of the C ionian scale, and, having the same notes, is related to C ionian. so, if you had the same C5 chord, you would play C ionian, D dorian, E phrygian, etc.
they are NOT the same scale though. the point of using modes in this way is to make a different note the ROOT note of the scale. (the root note is the 'home base', the note that is constantly being 'pulled' to).
It's not always easy to improvise this way and say 'ok, i'm playing in D dorian now'. its necesary to know the modes and their sound to know how to use them correctly.'
Hope that helps you undertand modes and thier application better! after a bit of 'hands on' experimentation you'll get the hang of it. Good luck!