Your second guitar chord is the chord of C. Like the G chord a simple version of this chord can be played with just one finger.
The C chord is played by pressing the 2nd string at the first fret. I recommend that you use the 1st, or index, finger for this. Start by putting your thumb on the back of the guitar's neck right near the head (the end of the neck). Then, slowly place your 1st finger on the 2nd string behind the 1st fret.
Don't press yet, just rest the tip of the finger on the string. Practice this movement a few times until you get comfortable with placing your finger tip on the string.
Now, try pressing down on the string. Keep the finger tip "upright" and ensure that it doesn't touch the other strings either side of the 2nd string. This is a little more tricky than for the G chord because there are two neighbouring strings. Practice pressing down a few times to get the hang of it.
On to the final stage, press down the 2nd string and strum the bottom four strings with the right hand. All four strings should sound clearly if you're pressing correctly. Practice strumming a while, but remember to release the string and relax your hand after a few strums.
Congratulate yourself, you're now on your way to playing C chords. Practice it for a few minutes each day and in no time you'll be playing it cleanly.