Well, if you don't press down hard with your fingers and just keep them resting on the strings then you will only mute and muffle the notes, and you will end up only hearing the open string notes.
When playing a chord like C (and I'm guessing you are playing it on a position lower down the frets) then you should press down hard enough so that the string is touching both frets at either side of your finger, and your fingertip just about touching the fretboard. You should hear a clear note. Do this with all the fingers and make sure they do not touch (even slightly) the strings you are not ment to be fretting.

I hope that makes it simpler to understand.

Your fingers will hurt at first, but soon they'll start to harden.

The best thing you can probably do is getting a teacher. He/she will answer any question you have and guide you properly through the great and mystic journey of playing a guitar.

And practice, practice, get naked and then practice!