1080i has vertical resolution of 1125 (1080 active) lines, so can show more detail.

But 720p has a faster frame rate (60 frames per sec vs 30 frames per sec), so it is better for fast-moving action, and is free of field artifacts.

Newer HDTVs are 1080p with 60 fps, so can display both formats as well as high-def DVDs from Blu-ray and HD-DVD players.

It is commonly stated that 1080i looks better up close, but from a normal viewing distance--about 6 feet from a 42-inch screen--a 1080p set won't look dramatically better than a good 720p TV.
The High Definition 1080p TV article at HomeTheaterHiFi.com explains the advantages of 1080p.

As of July, 2007

Type	Display Size	price *	weight (lbs.)
Flat Panel LCD	40-57"	47" $1,600-2,800	47" 55-65
Flat Panel Plasma	42-58"	50" $1,800-3,200	50" 85-105
Rear Projection DLP	50-65"	57" $1,800-2,700	57" 75-85
Rear Projection SXRD	50-70"	60" $2,700-4,000	60" 92-112
Rear Projection CRT		47" $1,900	47" 147
Front Projection	110-300"	1080 $4,000 720 - $1-2,000	15-24

* Price is for the specified size screen or resolution (for Front Projectors)
- 768p LCDs are available in smaller (20-42") sizes.
- Projection TV's are deeper (15-16" vs 4-5") so usually need to be floor standing.
- Front projectors require an external HD tuner, such as a cable or satellite box. For movies, you can hook up just a standard or high-definition DVD player ($250). - Front projectors are able to go up to 300"

720p - 720 lines of resolution progrssive (lines are painted in sequence)
1080i - 1080 lines of resolution interlaced (odd lines painted first 1/60 sec,
    even lines 1/60 sec.)
LCD - Liquid Crystal Display
DLP - Digital Light Processing
LCoS - Liquid crystal on silicon
SXRD - Silicon X-tal (Crystal) Reflective Display

Flat-panel TVs: LCD - Plasma:
LCDs (Liquid Crystal Display) use a matrix of crystals which act like a shutter, either allowing light to pass through or blocking the light.

Plasma TVs usa a matrix of gas-filled cells. A grid of tiny electrodes applies an electric current to the individual cells, causing the gas (a mix of neon and xenon) in the cells to ionize. This ionized gas (plasma) emits high-frequency UV rays, which stimulate the cells' phosphors, causing them to glow the desired color.

Plasma sets used to be the only flat-screen TVs larger than 40 inches, but LCDs now come in big sizes too. With the price gap between plasma TVs and big-screen LCD sets narrowing, more buyers are choosing LCDs.
Plasma sets have a wider viewing angle, slightly more lifelike color.
LCDs are generally brighter and less prone to glare than plasma and not prone to screen burn-in.
Plasma has a reputation for being less efficient, but tests at the Washington Post with a power meter didn't bear that out: The Panasonic Plasma used about the same amount of electricity over an hour as either LCD - though the Philips plasma drew about 25 percent more juice.

Projection (Front and Rear) TVs: DLP - SXRD - LCoS:
SXRD (Silicon X-tal (Crystal) Reflective Display) is a new technology from Sony is a refined variation of LCoS (Liquid Crystal on Silicon). SXRD is a "reflective" technology that sandwiches a layer of liquid crystal between a cover glass and a highly reflective mirror-like surface patterned with pixels. While LCD is a "transmissive" technology where light passes through the image chip Reflective displays such as LCoS and SXRD (and DLP) use the light from the TV's projection lamp more efficiently, resulting in higher picture brightness and contrast.Reflective displays such as LCoS and SXRD (and DLP) use the light from the TV's projection lamp more efficiently, resulting in higher picture brightness and contrast.

Trueness of color is also one of the significant differences between two major types of projectors: LCD, for "Liquid Crystal Display", or DLP, for "Digital Light Processing". DLP projectors tend to be very bright with higher contrast ratios, and they have become popular for smaller, lighter, "ultra-portable" projectors. To produce a color image, DLP projectors reflect a light through a rapidly spinning color wheel, which can lead to a slight flickering of color (called a "rainbow effect") and sometimes to an unnatural color appearance. Many newer DLP projectors now use faster-spinning color wheels, which can reduce the effect or 3 chips to eliminate the wheel.
LCD projectors often do slightly better with color, but they tend to shift the color a bit towards the blue range, and some significantly so.
DLP projectors also run very hot, which can be a problem for a portable projector in particular.
Due to the relatively high switching speed of DLP (up to 1,000 times faster than some competing technologies), DLP-based systems can more accurately display dynamic images (e.g. sporting events), while technologies with slower response times may produce some image smearing. For more information about DLP, see: www.ti.com/dlp.

LCoS "Liquid crystal on silicon" is a version of LCD technology, used by companies like Sony in high-end models. Compared to LCD, LCoS pixels are closer to each other, producing a smoother image and eliminating the screen door effect (a grid pattern on the screen) generated by LCD projection.
As of 2006, there was no consensus on which technology was best.

Bulbs on rear projectors have to be replaced on the average of every 6,000 hours while front projector bulbs last for 2-3,000 hours.

Front projectors can create an image up to 300" diagonally, howerver the larger the image the dimmer it will be. Consumer Reports found that 110" was the optimal size. For that size, you'd typically place a projector 10 to 15 feet away. The optimal viewing distance is 14 feet or so away from the screen -- about three times the height of the display area.
Image quality is just as good as that of plasmas and LCDs, However ambient light tends to wash out the picture.

You can get HMDI cables up to 39' (12 m.), longer cables may require a HMDI repeater.

Projectors require a screen. A 110-inch screen that has a matte-white viewing surface 4½ feet high by 8 feet wide will cost about $400. Or you can use a wall treated with special paint (such as Screen Goo). There are screens with more reflective finishes (called gain) designed to enhance brightness, but they might have a narrower viewing angle than a matte screen. So-called dark screens are designed to enhance contrast by improving black-level performance, but they see a small decrease in overall brightness.

Front projectors will also require a receiver/amplifier for the sound.

Connection Ports:
High-definition video comes in via two connectors: digital HDMI (high definition multimedia interface) and analog component video. Get a TV with two of each kind, plus a VGA (analog) or DVI (digital) port if a computer will be plugged in.
(You can get DVI to HDMI conversion cables/plugs.)

Reviews:
ConsumerReports _
cnet
PC Magazine

Links:
Audio-Video
Video Resolution
Home Theater and A/V at Crutchfield
HMDI cable benchmark
High Definition 1080p TV: Why You Should Be Concerned