An ASTC tuner is basically a built-in HD tuner. You need one if you want to watch TV without a external box. Cable Card is a thing you can get from your cable company that is a small card that slides into your TV, letting you watch channels on HD and premium channels with no need for a box. Few TV's have Cable Card support, though. HDMI and DVI, like YoYo said, are digital connections. HDMI has HDCP (copyright protection) built in, so be wary of that. Right now, theres a VERY small difference in quality between component and DVI connections, as far as I've noticed.