I think I see your confusion here -- or part of it. It's actually pretty easy.

Let's start with Dorian mode. It's actually D Dorian that plays notes from the C scale. (You said C Dorian.) So a D Dorian scale would be D-E-F-G-A-B-C-D. Another way to think of this is that a Dorian scale uses the same notes as the major scale a whole step below. Hopefully this is still making sense. :)

There are seven modes. One way to think about how these modes all work is to start with the progression of notes in a C major scale: C-D-E-F-G-A-B-C..., (and so forth, if you want to continue the scale up into the next octave). You can construct all seven possible modes of the C major scale, just by using the notes in a C scale, and starting on a different note. So, you have:

Ionian: C-D-E-F-G-A-B-C. This is the C Ionian mode -- a C major scale.

Dorian: D-E-F-G-A-B-C-D. You could think of this as the 2nd mode of the C major scale, since it starts on the second note of the C major scale, and proceeds up to the D one octave higher. Or, you could think of this as the D Dorian scale. Make sense?

Phrygian: E-F-G-A-B-C-D-E. (E Phrygian scale. Or, third mode of C major scale.)

Lydian: F-G-A-B-C-D-E-F. (F Lydian scale. Or, fourth mode of C major scale.)

Mixolydian: G-A-B-C-D-E-F-G. (G Mixolydian scale. Or, fifth mode of C major scale.)

Aeolian: A-B-C-D-E-F-G-A. (A Aeolian scale. Or, sixth mode of C major scale.)

Locrian: B-C-D-E-F-G-A-B. (B Locrian scale. Or, seventh mode of C major scale.)

Hopefully you're seeing a bit more clearly how this works. :)

You said you're not clear on how notes are flatted or sharpened in different modes. OK, let's take your example of a C Dorian that you linked in your post. You see, in the picture you linked in your post, how the 3rd and 7th notes (E and B) are flatted in the Dorian, compared to the naturals in a normal C scale? That makes it a C Dorian scale. But another way to think of this is that you've constructed a C Dorian, by using the notes from the major scale one whole step below. Compare the notes in a Bb major scale, with the notes the picture shows in the C Dorian scale:

Bb major scale -- Bb-C-D-Eb-F-G-A-Bb C Dorian scale -- C-D-Eb-F-G-A-Bb-C

See how the C Dorian just started with the second note of the Bb major scale, and ran through the same notes of that scale, until you got to the C one octave higher?

You can think about the other modes the same way. For example, take the G Mixolydian scale that I typed out (the fifth mode of the C major scale). Compare it to a regular old G major scale:

G Mixolydian -- G-A-B-C-D-E-F-G G major scale -- G-A-B-C-D-E-F#-G

So another way to think about the Mixolydian scale in any particular key, is to start with the major scale in that key, and flatten the 7th. So the F# of the G major scale becomes the F-natural of the G Mixolydian scale.

You can try the same exercise with the other modes -- compare the major scale with the corresponding modal scale, and see what you need to flatten (or sharpen) to turn one scale into the other.