MagSafe Charger not working
Have you tried resetting your SMC? Are you positive the power cord is working? Try different plugs, different laptops, different power cords. Test this thoroughly as you don’t want to go through the rest of this answer just to find out it was a bad cord/outlet. If nothing in the above works, keep reading.

Based on your symptoms, and assuming the second power adapter is functional, it sounds too me like a bad MagSafe DC-In board. This is the board that collects the power from the MagSafe cable and transfers it to the battery. I’d suggest 1) running Apple Hardware Test with the adapter plugged in to test this. If the power test comes back faulty, and you’re reasonably sure that the cable is working, the MagSafe board is your most likely problem. If you feel comfortable with DIY repairs on your laptop, check out this iFixit page where you can find the replacement part for ~$25 USD. If you don’t feel comfortable with this repair, take your computer in to an Apple Authorized Service Provider to get a quote for the repair. Good luck!

Please don’t hesitate to add comments/edits if you think this answer is wrong, incomplete or needs clarification.