Saving YouTube video's to your hard disc is not too difficult, you can use a firefox extension like VideoDownloader and a player like FLV Player or MPlayer to play the downloaded .flv files.
But on a low bandwidth connection, my interest was to save a video that you have just viewed in a browser window. I tried this on Firefox running under linux, but hopefully the method is similiar for other browsers and OS's.
Step 1: Identify the browser cache directory
Type about:cache in the address bar, and open the directory specified as "Cache Directory"
Step 2: Identify the video cache file
This can be tricky, but browse the above directory according the file modification date. The file you need should have some hexadecimal name (ex 55A2D2B9d01) and be of appropriate size (5-10 mb for most video's). Under linux, most file managers identify files based on metadata rather than extensions, so if FLV is a recognised type, your job is easy.
Step 3: Play the file
Rename it to .flv and use a media player like FLV Player or MPlayer