Weird battery issues There's this phone I got, it belonged to my girlfriend's mom, but she just gave it to me since it stopped working and I wanted it. The problem is it won't charge. I'll plug it in and the screen will turn on, but I get a picture of an empty battery with a yellow crossed out battery sign in the bottom right.

I've played around with it a bit and, after some testing, concluded that the problem is with a faulty battery - it won't charge in other devices, but other batteries will work and charge in this one. 

My question is, could this be something I'm able to fix easily, or would I have to buy a new battery? She said that it broke because someone tripped on the cable while the phone was charging and it fell on the floor. I don't really have enough interest in the device to buy a new battery, and I doubt she does either, so if I can't fix it that's it for the phone....
It depends on the battery, most are not repairable - if it's anything internal in the battery then there is nothing you can do but get a new one. Batteries have a limited lifespan anyway, it's normal to replace instead of 'fix'.

It is possible the connector broke off the batteries logic board (the part where the thing you plug into the device's metal makes contact with some other metal thing connected to the battery - part that isn't covered on the battery) has been shifted/broken off of what it's connected to and can't make a clean connection. If this is the case, you can try soldering the connector back onto the logic board.

Not knowing what phone type you have I can't give you a specific video that shows how to do this, but you can do a google search on your device/battery on how to connect the connector to the logic board - from there you'll know what to look for on the battery to see it it's not connected as well.