Open chords tend to sound "richer" than corresponding barre chords due to the fact that open strings resonate freely without any hindrance. Open chords typically sustain longer than their barre chord equivalent. An open chord is any chord that has one or more strings sounded without being fretted. Guitarists tend to use this name to refer to the chords that contain open strings that are played in the first position (first four frets).
Open Major chords
Open minor chords
Barre chords use a movable shape that can be shifted up in pitch and down in pitch along a string to give different keys.
Open seventh chords
Seventh chords, strictly speaking from a music theory perspective, means any seventh chord, including major, minor, dominant seventh, diminished and so on. In a beginning guitar context, however, "seventh chord" often means a dominant seventh chord, which is a major triad with a flatted seventh. The G dominant seventh chord, or G7 chord, is the notes G, B, D and F.
Chords in C major
Note that the chords in the key of C major consists of 3 major chords, 3 minor chords and 1 diminished chord. This holds true for all major keys.