The two main High Definition standards are 720p and 1080i.  Since there is plenty of information on the 2, I will not go into detail on them individually.  What there isn't alot of is objective, well-reasoned explanations of which one is better.  Mostly, this is because it is not a simple topic, and the winner depends largely on the context of the question.  Another reason this is so hotly debated is that people spend alot of money on HDTV's and that causes people to become zealots of whatever particular technology they've embraced. I'll try to compare the two within each of the important contexts, without personal bias.

Let's start with the fundamental question.  From a HDTV technology perspective, which is better?  The answer is, they are the same.  Before you leave a comment on how stupid I am, keep reading.  Fundamentally, we're talking about how much data can be displayed at a time.  For TV, that is limited far more by transmission bandwidth than anything else.  To maintain that bandwidth, the signal is compressed in a lossy fashion (see update below). In either format, you're going to get roughly the same "picture quality" watching TV.

Now, you're saying "shut-up, idiot.  One has got to be better.  Why won't my TV do 720p (or 1080i) if they are the same?". This is where the argument gets interesting.  The signal formats were created for "analog" TV technology, namely CRTs.  For CRTs, it's all about frequency.  (How many lines can you draw per second) 720p has more lines per second.  The problem is, you're still bandwidth limited, whatever that limit may be. Horizontal "resolution" comes down to bandwidth.  Any increase in bandwidth basically gives you more capacity for horizontal resolution, so again the two are the same.  The problem is, many HD sets are based on technologies with a "native resolution", meaning the TV is locked in to a certain resolution.  In these cases, the interlacing trick used by 1080i to gain temporal resolution at the cost of spatial resolution is useless.  [Added:] Many CRT-based HD sets cannot do 720p because it requires a higher horizontal refresh rate and the flyback circuit capable of 1080i is not capable of that speed (in short, you got screwed (myself included) to increase their profit margin). 1080i is refresh frequency comparable to 540p (if it existed as a standard).

So, the result is that most HD sets have a native resolution at (or near) 720p.  So, for those sets, a 720p signal matches the native resolution of the set and "looks" better.  There's alot of hype around 1080p sets, capable of displaying, you guessed it, a 1080p signal, which only really exists as the output of a PC at 1920x1080, which brings me to my next point.

Since basically all HD content is stored digitally, we're not just looking at lines of resolution, we've got frame sizes.  So a 720p frame is 1280x720, and you get 2 per second.  That's 1,843,200 pixels per second.  A 1080i frame is 1920x1080, but you only get 1 per second. That's 2,073,600 pixels per second.  So, compression and bandwidth aside, dealing with digital signal sources, you're getting more data (in the form of pixels per second) with 1080i than you are from 720p.  If that's the definition of better, then 1080i is better.

But, as I mentioned before for fixed resolution TV technology (anything but CRT), it takes a 1080p set to show 1080i in all it's glory.  Otherwise, 720p will probably look better due to the down-conversion resampling. Clear as mud, eh?  Well, I hope this clears it up for at least one person. [Added:] A signal will almost always look better in its native form.  Whenever there is a resampling step in up/down converting, you're going to have some degradation due to aliasing.

[Added:] I owned a 1080p DLP set for a few short days, and it was by far the prettiest thing I've seen.  it displayed both 1080i and 720p signals beautifully.  I sent it back, though.  All the upconverting in the video pipeline introduced a noticable delay that made playing video games very frustrating. :)

[UPDATE:] I had a comment asking about the lossy compression applied to video.  I'm referring to the digital compression that the video signal undergoes as part of the ATSC standard.  It uses MPEG2 compression (basically the same used for DVDs) to reduce the digital bandwidth (and therefore the analog bandwidth) that the video signal takes up so it fits in the "channel" (6MHz wide if I remember correctly).  This particular compression degrades picture quality (the amount can be controlled).  It is both a spatial and temporal compression, and gives very dramatic compression ratios.  Cable companies have more bandwidth to work with, but they usually pack the crap in as tight as they can rather than giving us better quality...jerks.

Note, there is also some "analog" compression going on as well to squeeze the ~20Mbit/sec digital signal into the 6Mhz channel.  The ATSC standard uses an 8-bit "vestigial side band" modulation (8VSB), while cable companies use a form of Quadrature Amplitude Modulation (64-QAM).  Both are beyond the scope of this article and neither of these come into play in picture quality as they are lossless forms of compression.

If you haven't guessed, I did a research paper on HDTV back in college.

[Yet another UPDATE:]  Want some hands-on proof?  Go download the 720p (146MB) and 1080i (211MB) versions of the King Kong movie trailer from Apple's site.  Compare the file sizes and then open them both, and you decide which one shows you more "data" (my definition of "better").