It's a little tricky. If you use Veoh or Realplayer, you should be able to download supported content and burn it to a disk. Then, if you rip it to windows media center, it should be in your computer...
You could use Keepvid (www.keepvid.com). You paste the link to the video into the box and then download it. Then you can choose the destination folder to save it into.
It depends where you're trying to download from. Do a quick search on the internet for every site you want to download from.
Example: download youtube videos
If you want to record your desktop/screen for playback, you will need a programme which records all the movement on your screen e.g. Camtasia Studio 7 (have to pay for it) or CamStudio (which is...