Extended Tertian Harmony
Around the turn of the 20th century composers began extending tertian harmony beyond the basic triads and 7th chords.  A 9th chord is created by adding an additional third on the top of a 7th chord.  An 11th chord adds a third on top of the 9th.  The 13th chord adds one more third.  An additional third (15th) would simply be two octaves above the root.

If the 9th is an octave plus a major second above the root, it is considered major.  If it is an octave and a minor second above the root, it is considered minor.  With 11ths and 13ths, the "major / minor" designations typically do not apply.  The 11ths and 13ths, if they are not part of an underlying tonal center, are generally labeled simply as # or b.  (See the examples below.)

The 9th chord is often seen as a dominant function (V9) with the 9th being resolved similar to the 7th, down by step.  When 11ths and 13ths are added, traditional resolutions and voice-leadings are no longer an issue.

These chords of extended tertian harmony can be found in the music of Debussy and Ravel, and have become part of the standard vocabulary of jazz.
Examples of 9th chords:

Often with extended chords, like the 11th and 13th, lower voices are omitted.
Examples of 11th and 13th chords:
Note:  When a 13th chord omits no pitches, as in the example above, all the pitches from the related scale are included.  In the E13 above, all the pitches from the related A major scale are included in the chord.