Barre Chords are played by barring your index finger across the fretboard to form a moveble chord pattern based off the 6th and 5th string more commonly but sometimes the 4th string is used.
These types of chords are common since they provide a simple reusable chord form that is easy to remember and once learned easy to use and play. They also allow guitarists to play chords up and down the neck using most of the strings for a fuller sound.
Root on the 6th String
The Root of the G Barre Chord is located on the 3rd fret of the 6th string.
Root on the 5th String
The Root of the C Barre Chord is located on the 3rd fret of the 5th string.
Root on the 4th String
The Root of the G Barre Chord is located on the 5th fret of the 4th string.
Notice that it is the same as the G chord with the Root off the 6th string but the 6th & 5th strings aren't played. Thus, you only need to bar the 1st & 2nd strings. It doesn't sound as full or powerful as the previous Barre chords though.