You can certainly have IE and Firefox on the same system. As a matter of fact, sometimes you will need IE for Microsfot things (but I hardly ever use my IE).
Remove any botched Firefox downloads from your desktop of wherever you saved them.
Use IE to download the Firefox installation to your desktop from here: http://www.mozilla.com/en-US/firefox/personal.html
Is says "Upgrade your Firefox" on the screen, but under that is the correct download button for a new installation.
That should put a Firefox Setup executable icon on your desktop. To avoid confusion or possible conflicts, I would close all the other open windows and double click the Firefox Setup executable on your desktop and the installation should start.
You need to verify that the download appeared to be successful, it is on your desktop and you double clicked the icon.
If you get an error, don't try to "describe" the error - post the exact contents of the error message as best you can.