Seventh, ninth, eleventh and thirteenth chords are all extended chords and are named so because they are extended with one or more thirds. The most common extended chord is a dominant seventh, in which a minor seventh is added to a major triad. A seventh dominant chord is extended with a minor third to build a ninth chord, a ninth chord is extended with a minor third to build an eleventh chord and, finally, an eleventh chord is extended with a major third to build a thirteenth chord.
See diagrams of extended chords:
C7 D7 E7 F7 G7 A7 B7
C9 D9 E9 F9 G9 A9 B9
C11 D11 E11 F11 G11 A11 B11
C13 D13 E13 F13 G13 A13 B13
We can compare the C7 chord with C9, C11 and C13:
C7: C – E – G – Bb
C9: C – E – G – Bb – D
C11: C – E – G – Bb – D – F
C13: C – E – G – Bb – D – F – A
So how do you play a chord with five or more notes? The answer is that you omit one or more notes and in many cases play the chord inverted (i.e. rearrange the order of notes in the chord). The alternative is to use both hands, but there is a risk for dissonance playing up to seven notes together.
Omitted notes in extended chords
7th: For the 7th chord it is not necessary to omit any notes, but it is possible to omit the fifth. A C7 with notes that could be omitted in parentheses would therefore look like: C – E – (G) – Bb
9th: The most standard procedure for the 9th chord is to omit the fifth. A C9 with notes that could be omitted in parentheses would therefore look like: C – E – (G) – Bb – D.
11th: Concerning the 11th chord the third is most often excluded. A C11 with notes that could be omitted in parentheses would therefore look like: C – (E) – G – Bb – D – F
13th: For the 13th chord the most common way is to exclude the fifth, ninth, and eleventh notes. A C13 with notes that could be omitted in parentheses would therefore look like: C – E – (G) – Bb – (D) – (F) – A.
More specific information and images can be found on the pages with chords for different notes, see links above.
Alternative chord symbols
Besides the most ordinary chord symbols alternatives can occur. C9 could also be written Cdom9. In the same style could C11 and C13 also be written Cdom11 and Cdom13 respectively.