They're the same thing (PVR = Personal Video Recorder).
Like Symm said, you'll need a fair amount of cash for a good capture card. Look at models by Hauppauge (HD PVR 2 Gaming Edition), AVerMedia (USB HD DVR/Dark Crystal) and BlackMagic (Intensity Shuttle) and see what you can afford/like the look of. Since you're gonna be streaming, you'll want to make sure the model you choose is supported by Xsplit (do a quick search on the Xsplit forums) or you'll need a powerful PC to stream in a different way.
If you have a PS2, get component cables and you should be able to stream record your games using your HD capture card too.
You don't have to, but some features are locked if you do like adding a game source. You can still stream off a capture card with a free Xsplit, but the quality will be hella bad.