I've embedded an mp4 video into my webpage. However, I first had to install Quicktime before I could see it. So of course, other users would have to do that also. Is there any way to play a video without installing the player if the user doesn't have it?
Do some research into HTML5 video. Unless the user is using a modern browser with video playback capabilities, they would need to have an app to provide video playback, as well as the codecs for the video file format that you will be serving them. At this point in time, MP4 is the "clear(est)" winner of the online video formats. However not all browsers provide built-in playback. For instance, Firefox currently does not support MP4. Future Firefox builds will rely on whatever video playback the native OS provides. So coming back to your question - no, there's no real way of guaranteeing video playback on the web. You do the best you can and provide links to users where they can download the appropriate codecs and players and call it a day.
Rather than hosting the video on your site's server, you might consider creating a YouTube or Vimeo channel, uploading the video(s) for your site there, and then embedding the appropriate YouTube/Vimeo code in an iframe on your HTML page. This is an especially good solution if your site will contain a lot of videos. If you only wish to use one or two, then go ahead and host them live if you like but be sure to load a video player plugin with the page. Just google web video player plugins, and you'll find an assortment. One of the ones Nancy O. has recommended is Pickle Player.