In Figure 1, the horizontal lines indicate the six strings of the guitar. The first string is at the top. When you hold your guitar in your left hand, with the strings facing toward the ceiling, this is what you see -- the first string will be facing away from you.
You can readily use the notation on my site if you're right-handed, and pick with your right hand and finger the strings with your left. If you're left-handed, the first string will still be facing away from you, but the diagram needs to be reversed from right to left. As usual, the lefties have to make an adjustment that the 'righties' don't need to. See Figure 2. Unfortunately, I haven't converted most of the pictures on this site to left-handed notation. Here's more information on left-handed guitar.
Figure 1. Right-Hand Chord Notation.
When the first string is played 'open' (string not held down), it produces a high-pitched 'E' note. As we go toward the bottom of Figure 1, the strings get thicker (on your guitar, not in this picture). The very bottom string is the sixth string. It produces a low-pitched 'E' note when played open. While that note is named the same as the open note on the first string, it is two octaves lower (as one drops an octave, the frequency of the note becomes half what it was before). The 'x' marks indicate which string, on which fret, is held down. The 'o' marks indicate an 'open' string. If a string is marked with neither an 'x' nor an 'o', it is not played. You can also see the standard notation for musical notes, along with the names of the notes (A, B, C, etc.)
Figure 2. Left-Hand Chord Notation.
The chord shown is A Major (usually referred to as A). The 'o' marks on the first and fifth strings indicates those strings are to be played open. The 'x' marks on the second, third, and fourth strings indicate they are to be held down (all on the second fret). You should use your second, third, and fourth fingers to hold down those strings when you play this particular chord (I don't indicate that on the picture). The sixth string is not played.
The names of the notes played are shown to the right of the diagram. You can also see their names, and standard musical notation. The open first string gives an 'E' note, the second string a 'C#', the third and fifth each give an 'A', and the fourth gives an 'E.' Note that one could also get an 'E' by playing the sixth string open. This is usually not done, since the bass (lowest-pitch, pronounced 'base') note of a chord should be, when possible, the primary note of the chord. For an A major chord, the primary note is A, given by the open fifth string. If you were to play the open sixth string, the chord would be denoted by A/E (A Major with E in the bass).
To understand how the chords are derived, go to the tutorial on keys. Table 3 is a quick summary of how chords are formed, and Table 2 contains all the notes in all of the (major) keys.
The 'A form' notation means this form can be moved along the guitar's neck. That is explained further in the Bar Chords section.