When playing a video on youtube.com a download button is added below the video.
Click it and a menu opens offering the available formats and resolutions of the video to download.
The available formats depend on the video.
Most videos are available in MP4 360p which is a good quality and can be played on most players including iPhone and other Smartphones.
Some videos are also available in MP4 720p which is HD resolution and better than DVD quality. I can also played on iPhone and other Smartphones.
There are also small download buttons in the search results of Youtube, Google and on Facebook. Click it to download the matching Youtube video in the best available quality