It's not a site which I use but a quick look shows that it uses Flash video, in the same way that YouTube does. In which case, you can save the videos (in FLV format) by using KeepVid:
http://tv-links.co.uk/link.do/1/4101/6178/4032 8/60296
Right-click on the link to the video on the TV-Links site. Select 'copy shortcut' (if you're using Internet Explorer) or 'copy link location' (if you're using Firefox).
In the KeepVid page, right-click in the box with the green frame. Select 'paste'. Click 'Download'. When the page refreshes, click the download link.
Choose where to save the file and replace the default name ('get_vid') with something more appropriate (e.g. 'tom&jerry01'). It helps to add the 'flv' file extension when you do so. (e.g. 'tom&jerry01.flv').
When you've saved your video, you'll need a suitable player. (Windows Media Player doesn't support the FLV format). The most popular free player is VLC:
http://www.videolan.org/
A good alternative is GOM:
http://www.gomplayer.com/
Chris