My laptop's Windows 7 installation is broken, so I can't boot Windows on it. I was thinking of installing Ubuntu to replace Windows, but I don't know how to do this. I am going to try installing Ubuntu 12.04.3. Can I do this with a 800 MB disc? Finally, is it possible if Windows is installed, but doesn't work?
Yes, you can make a bootable Ubuntu OS with 800mb disk. It doesn't matter whether your Windows installation works or not, you can replace windows with Ubuntu.
See this: How to replace Windows with Ubuntu?
As long as your BIOS or UEFI is still working (the splash screen with your computer manufacturer's name on it when your computer starts up), then you will be able to install Ubuntu.
Ubuntu 12.04, which is older than the latest version, 13.10, but has a longer support period. The latest version of Ubuntu, 13.10, has a shorter support period , but newer features.
Ubuntu 12.04 will fit on an 800 megabyte CD, but Ubuntu 13.04 will not. So, if you plan on installing Ubuntu 13.04, you'll need to use a DVD or USB.
If you would like to create a bootable USB drive (may not work on older computers), you should follow this guide for creating a bootable USB on Windows using Pen Drive Linux. Again, if you are running OS X, follow this guide instead.
In order to boot from your live CD/DVD or live USB, you'll need to press a key when your computer starts up. Assuming your computer didn't come with Windows 8 (in which case you should follow this answer for disabling Secure Boot), it will probablly be Esc or F2. When your computer starts up, look for some text on the screen that says 'Boot Menu' and a key on your keyboard (usually in the upper right-hand corner).
The process varies between computer, but there should be an option to select a CD/DVD drive or a USB storage device. If no USB option shows up (ensure that the USB drive is plugged into your computer before you start it), you'll need to use a CD or DVD instead.