Why does my macbook only turn on when the power adapter is plugged in?

I would try two things the first being resetting the System Management Controller:

1) If the computer is on, turn it off.
2) Disconnect the AC Adapter and remove the computer's battery.
3) Press and hold down the power button for 5 seconds and then release the button.
4) Reconnect the battery and AC Adapter.
5) Press the Power button to restart the computer.

If that doesn't work I would recalibrate the battery.

To calibrate the battery:

1) Plug in the MagSafe power adapter and fully charge the MacBook or MacBook Pro battery until the light on the MagSafe connector changes to green and the Battery icon in the menu bar indicates that the battery is fully charged.
2) Allow the battery to rest in the fully charged state for two hours or longer. You may use your computer during this time as long as the power adapter is plugged in.
3) Disconnect the power adapter with the computer on and start using it with battery power. When the battery's charge gets low, you’ll see the low battery warning dialog on the screen.
4) Continue to keep your computer turned on until it goes to sleep. Save your work and close all applications when the battery's charge gets low and before the computer goes to sleep.
5) Turn off the computer or allow it to sleep for five hours or longer.
6) Reconnect the power adapter and leave it connected until the battery is fully charged. You may use your computer during this time.

I've been having sort of the same problem. For the last two nights my power cord stopped charging. Tuesday night I worked off the battery and once it hit between 60%-65% my laptop just shutdown and it wouldn't restart. Last night the power cord actually smoked and melted. I worked off the battery and again it shutdown when it reached 62% and wouldn't turn on after that. I checked the system profiler and it says it's only at 32 cycles. I used my friend's power cord and then it turned on otherwise it wouldn't. I went to the Apple store tonight to get a new power cord. One girl I spoke to said the melting happens if you don't keep the cord straight, but I never ever curled it around anything. I then asked someone else if it was normal for the machine to shutdown after 60% battery power and the guy said no. He told me to bring it in to the Genius bar because it could be the battery. I just did the battery update so I wonder if it'll still happen. I'll make an appointment anyway just to be sure.

ususally its something to do with the battery however there is a battery update download you can try

http://support.apple.com/kb/HT1359

also try resetting your SMC:

turning computer off, removing battery and pressing and holding power button for 5 second

if that doesn't fix it, call AppleCare they can send you out a new a Battery

I am having this exact problem and I tried the update listed and taking the battery out like it says below. Did you end up getting a new battery?