In a nutshell:

Infinite kinds of chords, but I'll be psycho and try and classify barre chords into 3 types:

E Shaped
A Shaped
C Shaped

Take E Shaped for example. You play an Em "shaped" chord with your first finger barring the 2nd fret...what are you playing? F#min. Because the root (E) was moved up 2 half steps (or one step) to an F#.

Take A for example. You play an Am "shaped" cord with your 2nd, 3rd, and 4th fingers, using your 1st finger to barre the 5th fret. 5 half steps up from A is...
A
A# 1
B 2
C 3
C# 4
D 5

Hence, Dmin.

Sorry to cram it into to a couple short paragraphs, but I hope this helps.