Here are the formats you can download using this script (Wikipedia has the full list):
MP4 360p (384x288, 480x360, 640x360) - all videos are available in this format. Use it if you want to play videos on a mobile phone (iPhone, Android) or to save space.
MP4 720p (1280x720) - HD format, available for an increasing number of videos. Use this format if you want to play videos on a tablet (iPad, Motorola Xoom, Samsung Galaxy Tab) or a laptop.
480p and 1080p are no longer available as standalone videos downloaded from youtube.com. YouTube switched to adaptive streams and uses separate files for video and audio, which need to be merged to get a video file. This is more complicated and requires more advanced tools (ffmpeg), so I haven't added them yet.
After downloading videos, you can play them using free applications like:
VLC (Windows, Mac, Linux / plays both MP4 and FLV videos) - recommended video player
video players for mobile devices (for example, MP4 360p videos are better suited for older smartphones like iPhone 3GS, HTC Desire, Galaxy S/S2; MP4 720p videos are great for more recent smartphones (iPhone 4/4S/5, Galaxy S3), 7-inch tablets and older 10-inch tablets (iPad 1/2), while 1080p videos work better on the latest iPad, Nexus 10 and other high-resolution tablets and smartphones.)
Automatic updates: Scriptish and Greasemonkey for Firefox and Tampermonkey for Chrome support automatic updating.