A sixth chord is any triad with an added sixth above the root.   For example, a major sixth chord built on C (denoted by C6, or CM6) consists of the notes C, E, G, and the added major sixth.  Here are the 5 shapes for the sixth chord found in the open position.  Other sixth chords (such as F6 and B6) can only be played by using a barred form of one of these 5 shapes.