Barre chords are simply chords played on the fretboard, usually without open strings. In other words, the same fingering for an E Major (0-2-2-3-0-0) can be moved up the fretboard to the second fret. Instead of the open E and B strings, they are barred and the result is an F♯ Major (2-4-4-3-2-2). If you were to move this formation up the eighth fret, the result would be a C Major (8-10-10-9-8-8).
They may occur anywhere up or down the fretboard, so long as the prescribed fingering remains the same. These chords are named for the bass note, so whichever note is the lowest is the name of that particular chord. For example, if you were to bar the A, D, G, and B strings on the third fret, that chord would be a C9sus4; if you were to bar the sixth fret it would be a D♯9sus4 (or an E♭sus4 depending on the key of the song).
These chords make playing an E♭ or A♭ Major possible while remaining in natural tuning (e.g. E♭ x-6-8-8-8-6 and A♭ 4-6-6-5-4-4). Otherwise, these chords could not be played unless the tune is altered or a capo is used.