There are two kinds of chords here. The first are open-position
chords that rely on open strings. As such, they're not moveable. For
example, you can't take an open-position D Major chord and slide it
up the neck two frets to make an E Major.
The open position chords are a good place to start. There are chords
for six keys which, if you master them all, gives you the ability to
play a song in a key that will keep the vocalist happy. The circles
show you where your fingers go; the white circles indicate the root
note of the chord. Circles behind the nut indicate an open string.
The numbers tell you what finger to use (1=index, 4=pinky). An
'X' behind the nut indicates the string is not
to be played.
Once you've mastered the open-position chords, the next step is the
moveable chord forms. These use no open strings so they can be
moved up or down the neck as your please. The root note of the chord
is important here because it guides you to the fret where the chord
should be played to give it the tonality you want. This is not an
exhaustive list but it covers the most common chord shapes.
Example: You want to play an E-flat major chord. Find one of the
moveable major forms and locate it on the fingerboard such that the
root note (the white circle) is on an E-flat note.