You downloaded the ISO image. Then you double-clicked on it:

When the downloads are complete nothing else happens, but the file
"ubuntu-12.10-desktop-i386" can be found through explorer. Double
clicking on this file opens a dialogue box called "Windows disc image
burner" which gives the message "a disc burner wasn't found".

Don't double-click on it. Given that you've successfully downloaded the ISO image:

To Install a Normal (non-Wubi) Ubuntu System Alongside Windows, via USB

1. Run UNetbootin or Universal USB Installer to install the ISO to the USB flash drive. Whichever you use, you'll run the program, then in the program you'll fill in some information, including telling it where to find the ISO image you've downloaded. Double-clicking on the ISO itself will not work.

   
   



2. If your backups of documents and other important files are not current, update them. (Or if you don't have any backups--I've found people occasionally don't--make one.) There is a small but significant chance of data loss when modifying your partition layout (which happens as part of installation).

   
   



3. Reboot, with the USB flash drive attached. Make sure your Windows system shuts down completely and cleanly. In particular, if it crashed, boot back into it and let it try to fix whatever might be wrong. (Unless it always crashes and never shuts down cleanly, in which case there's nothing you can do.) Make sure you're rebooting or shutting down...not hibernating.

   
   



4. Depending on how your BIOS is configured, when your computer starts it might automatically boot from the USB drive, or you might have to tell it to do so by bringing up the BIOS boot menu. They key to press for this varies depending on manufacturer. Dells use F12. Usually your computer tells you, at least briefly, what key to press to access the boot menu.

   
   



5. Select Try Ubuntu (if you want to test out Ubuntu from the live USB first, which I recommend) or Install Ubuntu. If you select Try Ubuntu, you can install it by double-clicking the Install Ubuntu icon on the desktop in the live environment.

   
   



6. When the installer runs, tell it to install Ubuntu alongside existing operating systems. This will shrink down your Windows partition and create partitions for Ubuntu. (It is usually able to do this; making sure Windows shut down cleanly increases the odds of success. If it doesn't work, you may have to go back into Windows and turn off swap and hibernation. You can reenable them after installing Ubuntu, and you should, at least the swap. I don't recommend turning these things off at all unless you need to.)

   
   

   Do not tell it to install using the entire disk, unless you want to overwrite the Windows system (destroying it and data inside it--it's very hard and often impossible to recover the old data in this situation).

   
   

   I recommend keeping the Windows system for now. Having it doesn't obligate you to use it, and even if you know you prefer Ubuntu, having the old OS can be a valuable "safety net." (If your Windows system were seriously broken, such that it could not really be used for anything, then I would not be advising this.)

   
   



7. Assuming everything worked correctly, when you reboot, you will be able to choose between Ubuntu and Windows. Both should work.

   
   

TL;DR:

Don't double-click the ISO. Open it in UNetbootin or Universal USB Installer, as any instructions for installing Ubuntu via USB tell to do. Then boot from the USB flash drive and you can install Ubuntu.