if you want to share your PS2 files all you need is the sharkport
if you want to play PS2 games over the internet you are going to need the network adapter for the PS2. good luck finding one though as they sell out quickly.
To hook up your PS2 to your network so you can use your cable connection you are going to need to buy a network hub. Connect the hub to the computer, then the Cable moem to your hub and the PS2 to the hub, this way the internet connection is be shared between your PS2 and computer and you can now play PS2 games online if they support online play.
Jus as Tool say, if you want to download or upload PS2 saves, then you'll need a SharkPort or Xport.
If you want to play PS2 games on your PC (monitor), then you need a VGA Box/Adapter.
If you want to capture videos, then you need a video capture card.
Check out my old and abandoned SOCOM website (my first html btw, so don't hate if it's cr@py): www.geocities.com/marjoh1215 for some network info and setup (under the news section I think).
But I think it's better to post this question @ gamefaqs.com (PS2 section). You'll get quicker and more responds there.