Modes are alternative tonalities (scales) that can be derived from the familiar major scale by starting on a different scale tone. Music that uses the traditional major scale can be said to be in the Ionian mode. For example, in the key of C, the Ionian scale would be C D E F G A B.
The other familiar mode is Aeolian. It can be derived by starting the Ionian (major) scale on the sixth scale tone. For example, A Aeolian would be A B C D E F G. This is the A natural minor scale.
The other modes can be derived similarly, by starting the major scale on the other tones. If we stick with only the white notes on the piano, we can derive seven different modes. These are:
As you can see, each mode denotes a unique set of intervals above the root tone. The result of this is that music written in each mode has a very distinct sound. Progressions that sound familiar in one mode may sound otherworldly in another mode.
The tonic chord alone sounds different in many modes. In Ionian, Lydian and Mixolydian, the tonic triad is major. In Dorian, Phrygian and Aeolian, the tonic is minor. In Locrian, the tonic is diminished. Because the diminished triad sounds so unstable, Locrian is the most rarely employed mode.
However, since the modes all contain the same set of notes, any or all of them can be used within a single piece without accidentals. A certain mode can be achieved simply by creating a section which resolves to or is based on the tonic of that mode. Again using the white keys of the piano to illustrate, a section resolving to A would be in the A Aeolian mode, while one resolving to C would be in the C Ionian mode.
The modes can also be written by noting their differences from the Ionian, i.e. what sharpened or flattened notes they contain. This approach yields the following (in order of increasing number of flats):
Thus the E Phrygian is an E Ionian (major) scale with flattened second, third, sixth, and seventh degrees. It can be observed that, for a constant tonic, each mode contains the same notes as the previous except for one note being lowered, and that the lowered notes appear in the following order: IV (natural, lowered from IV♯), ♭VII, ♭III, ♭VI, ♭II, ♭V. This is known as the "order of flats", and can be used as an aid for memorizing the modes.