Earlier today when browsing through a university course site, I found that the site was not properly shown in Firefox browser. There was a tiny message at the bottom of the page saying that site was best viewed in Internet Explorer. So I had to open Internet Explorer and view that site in it. But if you do not want to bother opening multiple browsers just to view a site properly, then you can use the IE Tab Plus addon for Firefox. IE Tab enables you to use the embedded IE engine within tabs of Mozilla Firefox so that you can view IE compatible sites right inside Firefox.
You can visit the Mozilla add-on website to install this IE Tab Plus add-on. It requires restarting your web browser. After the installation and restarting Firefox, IE Tab Plus would be ready to use. You can see the IE Tab Plus icon in the add-ons bar at the bottom-right corner of the Firefox window. You can left-click on this icon to switch the rendering engine between Firefox and IE. Alternatively, you can also right-click on this icon and select Switch Rendering Engine. This would open the current website in the alternate browser engine.
In the IE Tab Plus settings, you can add some sites to be always opened in Internet Explorer. For example, Microsoft sites like windows.update.com require to be opened only in Internet Explorer and you can set it to open open always in Internet Explorer. To open the IE Tab Plus settings, right-click on the IE Tab Plus icon in the addons bar and select Settings. Then switch to the Sites Filter tab and type in the site address in the URL field. Finally click on the Add button.
You can add sites using their full URL or using regular expressions to match a wide pattern of sites. The IE Tab Plus addon already has some sites added including the MHT and MHTML files (web archive files) for which Firefox has no native support. These files would be opened in Firefox using the IE rendering engine. You can also open the MHT/MHTML files using the unMHT addon for Firefox.
So if you want to open sites, that are designed to work only in Internet Explorer, in Firefox, then you can just install the IE Tab Plus add-on for Firefox. With the add-on IE Tab Plus for the Firefox browser you can view Internet Explorer compatible websites right inside Firefox. You can install it from the Firefox add-ons page at https://addons.mozilla.org/.