[cpinegar]

Very good question.

Assuming you are playing in the key of C/Am, then the I chord is C major, and the ii chord is Dm or Dm7, etc.

Dorian fits perfectly on top of a ii chord, because it is the ii scale. If making chords and scales get along together without any tonal tension is your goal, then you may opt to use modes to "fit" the chords and scales together in this way. This is one valid way to use modes, which you can think of as just playing the notes in C/Am, but your emphasis is stronger over chord changes to and from Dm, instead of C or Am. To answer your specific question, when you are playing the key of C/Am, the notes your ear hears are all the same, but emphasis delivered by the bass or rhythm guitar sections push the tonal center up and down along the harmonic scale. If your goal is to follow the tonal center changes with your melody, and resolve from ii to I, then your scale should also shift its emphasis to end on the I (most commonly down through the ii or up through the VII scale degree).

Another way to think of modes is as substitutions for the classic Major or Minor scales.

Major modes (by virtue of the major 3rd) are:
Ionian (major)
Lydian
Mixolydian

Minor modes (by virtue of the minor 3rd) are:
Dorian
Phrygian
Aeolian (minor)
Locrian

For example, where you would normally use a minor scale over a vi chord, experiment with a Dorian, Phygian or Locrian instead. The root, minor 3rd will keep the vi chord linked to any of these scales, but the linkage is a little looser with the other degrees, and the emotional result can be one of excitement.

It takes time and practice to know when to think of a mode as a shift within a key, or as a substitute for a major or minor scale, but practicing both gives you a more seasoned variety of options for your listeners.