A commercial answer to the problem would be to use a service like Ooyala. They have a fairly comprehensive analytics portal that's a part of their product. They license and change on a per-stream-served basis.
If you're looking for something a little less pricey and like to customize, I suggest extending your video player using Google Analytics Events. The event framework allows you to specify a category, an event name, a string, and a number. This covers most needs.
If you are looking for the easiest possible answer, just upload the video to youtube. When you look at it under "My Videos", there will be a button next to the video labelled "insights". Insights has five sections: "Views", "Discovery", "Demographics", "Community", and "Hotspots". Hotspots is probably what you're looking for. You can then simply hand the embed code to anyone that needs to play your video.