The Interval between notes is based on the number of semitones each note is from its predecessor.
Here is a typical octave of modern equally spaced semitones based on a tonic (starting note) of C for reference:
Use this table to construct the mode starting on any note, using the pattern of intervals below.
If we look at the intervals of notes which make up the Ionian Mode (which starts on C), we get:
This pattern of intervals: T-T-S-T-T-T-S* is the characteristic of the Ionian mode, and gave us (or was adopted as) the modern Major Scale. Because modes are characterised by its interval structure, you can start on any note and progress with the same intervals to produce an Ionian mode in that 'key'. This would be 'D Ionian':
You can see that using the Ionian Mode intervals, which we said gave us the modern major scale, the Ionian Mode starting on 'D' is identical to the modern key of D Major.
* T = Whole Tone, S = Semitone