Hz for HDTVs usually is in reference to an LCD HDTV's refresh rate. This is how often/quickly its refreshing (re-drawing) the screen.
With fast moving motion, the "standard" of 60Hz is found to not hold up very well. Basically consumers found out at LCDs are inherently bad with fast moving images. They suffer from motion blur (ghost trails) at 60Hz. So that's when you started to get these TVs with 120Hz or 240Hz. This was the manufacturer's response to the problem. They artificially have the TV draw extra frames (either by double or interpolating) to mask the motion blur. But right now you have to pay a huge price premium for this.
This only occurs with LCDs. CRT, Plasma, and DLP all have near instantaneous response times. They do not suffer from motion blur at all. Sometimes their spec sheets list refresh rates, but that's just marketing, it really doesn't mean much. The technologies are inherently already great.
So the problem has nothing to do with HDTV, it only has to do with the technologies that playback HDTV. There is no motion blur or skipping of pictures on the raw original HD materials. Its purely only on playback on LCDs.
Also, TV size matters. The larger your TV the more likely you'll notice this. But below some sizes you might not see it at all, even with an LCD. Sets of 40" and higher, and moreso 50" and higher would definitely exhibit the motion blur problems with the standard 60Hz set. But sets of 37" and smaller, not so much, if at all. So you could actually get one of those at 60Hz and still have an amazing HD experience (on an LCD).