A 6/9 respectively 6/9 chord is a 6th chord with an additional major 2nd/9th. The major 9 is the same tone as a major 2nd, but since it is a chord extension (the 2nd is not explicitly replacing the 3rd as sus2), it is written as 9 in the chord symbol. On the fretboard you can find it a whole-step (2 frets) above each root tone.
The C6(9) respectively C6/9 chord has the notes C (1), E (3), G (5), A (6) and D (9). The 5th might be omitted, but for technical reasons the 6/9 chord is rather played without the 3rd on the guitar. Strictly speaking this would be a C6sus2 chord, but that is uncommon, because you'll probably never completely ban the 3rd here.
The most common application of a 6/9 chord is the use as final chord of a song. It can be added to a final chord, for examle: Dm7 G7(b9) Cmaj7 C6/9
or replace the final chord, for example: Dm7 Gsus9 C6/9
Following a consistent syntax where chord extensions are written in brackets, the 6/9 chord would be written 6(9). However, the notation 6/9 with a slash has established itself in this particular case, but this one has nothing to do with the so-called slash chords.