Godaddy's shared hosting control panel does not include a tool to change or add Mime Types, so out of the box, your hosted site will probably not be able to serve .webm, .ogg, .ogv, or .oga content. In addition, by default, Godaddy servers are configured to use the mime type "video/mpeg" for the .mp4 extension rather than "video/mp4", which is why your mp4 video might be failing to play even in IE 10. However, if your site is hosted on IIS 7+ you can add some entries to your web.config file. Find the <system.webServer> element and look for a section called <staticContent>. If it is not there you should add it so that it looks something like this:
<!-- Other elements -->
<mimeMap fileExtension=".m4v" mimeType="video/m4v" />
<mimeMap fileExtension=".ogg" mimeType="audio/ogg" />
<mimeMap fileExtension=".oga" mimeType="audio/ogg" />
<mimeMap fileExtension=".ogv" mimeType="video/ogg" />
<mimeMap fileExtension=".webm" mimeType="video/webm"/>
<remove fileExtension=".mp4" />
<mimeMap fileExtension=".mp4" mimeType="video/mp4" />
Note that because the .mp4 extension already has a mime type configured for it on the server, we have to remove it first.
Troubleshooting: If, after adding this section you start seeing Error 500 server errors on your site, first make certain that there aren't now 2 staticContent sections in your system.webServer section, then remove all of the mimeMap elements and see if the error goes away. If it does, simply add them back one at a time until you cause the error again. Add a remove element before the mimeMap(s) that cause the exception.