The browser used must support HTML 5 video, most modern browsers do.
HTML 5 video is now a standard, and easier to embed in a website. Plug-ins like Adobe Flash, which has problems with Apple products, are not required. Videos may be shown as a full screen background like the one showing on the section here, or used as a normal element like the video below (Big Buck Bunny).
Some website designers don’t like to use videos as a background. Their reasoning is that it is a distraction, taking the visitors attention away from the real reason for the page. This can certainly be true. It also may not be true, but it is true that just because we can do somethings, doesn’t necessarily mean we should do them.
It is the website developers job to point out the pros and cons so that the site owner may make an informed decision.