Things get more complicated from here. The battery on the computer was held in place with packing tape prior to my beginning the upgrade. The battery would not come out - meaning I can't access the RAM. I tried to force the battery with a knife but a plastic backing on the battery begin to peel off.
Did you tape it down or was there a previous owner? I would look around for videos online about battery replacements on your model, and they should help identify the tabs Greg mentioned. I'm not sure if these can be replaced, but if they are cheap, it will be a lot better than packing tape.
If you think you might have badly damaged the battery, you might want to take it in to a general computer repair shop and ask if it's safe to use in its current state. Someone should be able to look at the battery for 30 seconds and tell if it's good or not for free. If you started to peel open the battery casing, I would be nervous to use that long term.
I replaced the battery on my 2008 macbook, but those plug and play from outside of the computer, so I can't really help you with specifics. If you decide you need a new battery, I would highly recommend getting a 3rd party battery. Mine was much cheaper than the Apple replacement and has been working great. The one I got on Amazon was ~$30, it looks like yours run around $45.
Good luck!
Edit: I forgot to mention that if you have your original Apple battery are having any of the classic macbook battery issues (not charging correctly, poor battery condition in the top menu bar, etc) it's worth considering just replacing the battery now. You didn't mention this happening, but it's so common on older macbooks I thought I would mention it. This is why I had to replace my battery-- I had warning messages and then it stopped charging suddenly.