Your first quote was confusing because it was self-contradictory. So I'll try to explain it again.
A new HDTV today is either 720p or 1080p (some CRTs are 1080i, but I don't think anyone here is thinking of getting a CRT, so we'll ignore that). But a DVD only contains info at 480i. So to show the DVD on the TV, you need to convert the info from 480i to 720p/1080p. All TVs are capable of doing this conversion. However, some people believe that the scaler in certain DVD players do a better job at converting 480i to 720/1080p than the TV does.
As I said, some people see the difference, some don't. Personally, I've never done the comparison.
As for a special cable for upconverting players, you either need component cables or HDMI cables, which aren't really "special."