Why won't my batter charge?

I got a macbook air from a friend, because the logic board was supposedly fried. however, I opened it up and a cable wasn't plugged in. I plugged the cable in and closed the machine, and the computer worked fine, the thing was that it needed to be plugged into turn on. I took the macbook air, to west world (an authorized service provider) and they did a scan of the computer to see what was wrong. They said it was the battery, so I ordered one from here (ifixit.com). The battery came fast and I installed it. now heres the puzzle.

With the new battery in the computer runs, but won't charge.

With the old battery in the computer runs, but needs to be connected to power or it won't turn on... me and my dad made a video that explains it better:

http://youtu.be/tAbs-0Wf0OE