The little red light is the digital audio signal, there is a little flap inside that when pushed open allows for digital audio instead of through the speakers, head phones work because they are making a connection and cutting off the digital audio.
Pretty easy fix, you have to close that little flap. You can do it either by plugging and unplugging a TOSlink cable (if you have one around), this should dislodge the flap, you could take it into an Apple Store and they could do it, or you could find a little round plastic (or a toothpick) and go in and see if you can get the flap to close by pushing on it.
Hope this helps.