Hi Elizabeth, good question! Each mode can be thought of starting on a different pitch of our major scale. For now let's keep it simple and stay in C Major.
Ionian - Starts on scale degree 1. C D E F G A B C. Identical to our C major scale.
Dorian - Starts on scale degree 2. D E F G A B C D. Similar to a natural D minor scale but with a raised 6th.
Phrygian - Starts on scale degree 3. E F G A B C D E. Similar to a natural E minor scale but with a lowered 2nd.
Lydian - Starts on scale degree 4. F G A B C D E F. Similar to a F major scale but with a raised fourth.
Mixolydian - Starts on scale degree 5. G A B C D E F G. Similar to a G major scale but with a lowered seventh.
Aeolian - Starts on scale degree 6. A B C D E F G A. Same as our A natural minor scale.
Locrian - Starts on scale degree 7. B C D E F G A B. This is a weird one. It's almost never used, but we still need to know it. It's really just a scale built around a diminished triad (B D F).
So back to your question: if you wanted to write a Lydian mode based off of a D Major scale, you would simply write a D Major scale but start on the fourth scale degree (G A B C# D E F# G).
It's also important when analyzing music to be able to identify which mode is being used. Let's say that you're assigned a piece of music that was composed with the scale E F# G# A B C# D E. What mode would you say that this piece is in? To figure that out, we would first need to identify the key signature. I see three sharps (F# G# and C#). I know this is the key signature for A major. I also already know my starting note is E. Because E is the fifth scale degree of A Major, and our Mixolydian mode begins on the fifth scale degree, that would mean the piece is written in E Mixolydian.
To recap:
1) Identify the key signature
2) Identify the starting note of the mode
3) Figure out what scale degree is the starting note of the mode
I hope this helps!
Johan