How to work out the Ionian, Dorian, Phrygian, Lydian, Mixolydian, Aeolian and Locrian modes on any starting note.
The following pictures show the most simple form of every mode of the major scale, i.e. every mode when they are played on only the white keys on the piano. You can find more details about these modes as well as the modes of the harmonic and melodic minor scales here: Complete guide to all the modes of the major, melodic minor and harmonic minor scales.
Each different mode can be worked out by playing a C major scale but starting and finishing on a different note. I.e. if you play a C major scale but start and finish on D, you are actually playing a Dorian scale, which is the second mode of the major scale. If you play a C major scale but start and finish on E, you are actually playing a Phrygian scale, which is the third mode of the major scale. It is the sequence of tones and semitones* that gives each scale its character. The following pictures show the specific sequence of tones and semitones for each mode. If a piece of music is written in E phrygian it will not sound like C major even though the key signature (and lack of sharps and flats) will be the same. This is because the sequence of notes and chords will be different, and the tonal centre will be E rather than C.
You can transpose all of these scales into ones which start on different notes just by following the correct sequence of tones and semitones. Jazz/pop music often uses versions of the Dorian, Phrygian, Lydian and Mixolydian scales, but are not limited to only these. Classical music most frequently uses the Ionian (major) and minor scales, but also features other modes, such as the Phrygian.
Ionian (same as major scale)

You can work out a major scale (or ionian scale) starting on any note by following the sequence: tone, tone, semitone, tone, tone, tone, semitone.

You can work out a dorian scale starting on any note by following the sequence: T S T T T S T. (T = tone, S = semitone.)
Phrygian

You can work out a phrygian scale starting on any note by following the sequence: S T T T S T T.
Lydian

You can work out a Lydian scale starting on any note by following the sequence: T T T S T T S.
Mixolydian

A mixolydian scale starting on any note follows the sequence: T T S T T S T.
Aeolian (same as natural minor scale)

An aeolian scale, or natural minor scale starting on any note follows the sequence: T S T T S T T
Locrian

A locrian scale starting on any note follows the sequence: S T T S T T T.
You can download a free PDF showing all of the above modes and their corresponding sequences of tones and semitones:
* A semitone is the smallest distance from one key to the next on the piano. E.g. a C to a C# would be a semitone, because there are no other keys in between these two. A tone (also called a whole tone), is two semitones.
You can listen to some of my modal improvisations here.