In my own words, of course - there's probably a lot more on Wikipedia, other sites, not to mention the preachings of home theater enthusiasts (aka DVD snobs) such as myself. I'm tired of educating boneheads on sites populated by militant movie buffs living in their parents' basements downloading scripts and sharing insider information about people they claim to despise yet can't stop discussing... Oh wait, that was Affleck's line in "Jay and Silent Bob Strike Back", wasn't it? Yeah... Now then...
Lesson One: How movies are shot (filmed)
Most movies are filmed using a wide-angle lens. There are several choices to choose from, one of the more common being 16:9 (meaning the frame is not quite as wide as it is tall, but close). Some are wider still, like you may see 2.39:1 or 2.85:1, meaning it's much wider than it is tall. However, most television programs are shot with a 4:3 lens - you guessed it, almost square. Nowadays, television programs are starting to be shot at 16:10, but I'll get to that in Lesson Three. There isn't much else to cover except to go over the various aspect ratios. As to the why, it's up to the director and/or producers. Human vision is naturally wide-angled, but most TVs are 4:3, so it's a balance of convenience and staying natural.
Lesson Two: DVD Choices (more than you think)
You probably know about Fullscreen and Widescreen, but within those categories, there are a couple subchoices. Fullscreen is a misnomer, assuming your screen is 4:3, and refers to the fact that the image will most likely fill your screen, corner to corner, top to bottom, left to right. And widescreen is also a misnomer, being that it implies it fills a wide TV, when the reality is that, unless the movie is anamorphic or 16:10, you still get black bars (albeit smaller).
Fullscreen is usually substituted for Pan & Scan, which is what they were called in the VHS days. Since a fullscreen transfer loses about a third of the picture, the transfer process involves choosing what will be cut. Mostly it's extra landscape that isn't needed, but sometimes there is important visual data on both edges. A pan is when the camera slides to the side or up and down in a straight line, but in this case it's when the focus shifts from one edge of the frame to the other. Sometimes you can tell - I can. It'll seem unnaturally smooth. The camera isn't moving per se, but you go from, for example, looking at the very right, to looking at the very left.
Then there's letterbox, and it's important to differentiate widescreen from letterbox. On widescreen, there are no black bars - the black bars you see is just where there is no visual data. A letterboxed movie uses 4:3 frames, and the black bars are added to force the director's original vision onto a 4:3 TV. A proper widescreen video on a fullscreen TV will often stretch vertially to fill the TV, depending on the TV and DVD player. Letterbox is fine if you have a 4:3 TV, but it's terrible on a widescreen TV, because you get the black bars all around.
Within widescreen, you have all the different aspect ratios, of course. The main choice here is Anamorphic, which means it'll stretch to fit the screen. Apparently, this is a good thing, as opposed to a widescreen DVD stretching to fill a TV. I actually don't understand this, and Wikipedia is a little too technical on the issue.
So you have Letterboxed Fullscreen (Rare, avoid this), Pan & Scan Fullscreen (common, second choice), and Widescreen (best choice).
Lesson Three: Your TV, and choosing the best DVDs
As I understand it, there are two kinds of TVs as far as aspect ratio goes. 4:3 is your average TV. At least 90% of televisions are 4:3. Your grandfather's TV was 4:3. The TV in your work's break room is probably 4:3. With rare exception, most TVs most people will have will be 4:3. Within 4:3, there are two kinds - round and flat. This refers to the front surface of the tube. If the face is flat, resolution will be much higher. If you must get a 4:3 TV, make sure it's flat (this doesn't mean LCD/Plasma). Usually these are CRT - tube based and heavy - but there are a few LCD ones, and the big projection TVs that were hot in the 80s.
Then there are 16:10 TVs, and they come in a few flavors of their own. They actually make CRT wide TVs, and they are comparable in price to the 4:3 TVs. LCD is the base flat kind, and then Plasma if you have the money. These are what home theater enthusiasts have or want.
If you have a 4:3 TV and 1) you don't plan on getting a 16:10 TV and 2) you don't really care about that extra third of the frame in your movies, Fullscreen Pan & Scan is your best choice. If you plan on getting a 16:10 TV in the future, you should probably select Widescreen. Fullscreen movies look bad on 16:10 TVs. Instead of black bars on top and bottom with widescreen, they're on the sides, and it's much worse. (As I said, the human eyes naturally see "wide", so it obscenely looks as though something is missing.)
Most TV programs are shot in 4:3, or at least they used to be. Some still are, but some now (and more later) are shot in an anamorphic ratio that almost fills the screen on a 4:3 TV (you might notice real small black bars) and fills a widescreen TV perfectly. This format will probably take over. I'm not even sure if HD-DVD and Blu-Ray (DVD's would-be succeesors) even come in fullscreen. We could be looking at a future (say, 5 years down the road) where everything is simply widescreen, except for some "archaic" movies and TV sets.
Stanley Kubrick was known for shooting movies in 4:3. Full Metal Jacket certainly is; as for the others, I think some are and some aren't. But if you have a wide TV, there's almost no getting around buying movies or TV shows in the format they were shot in - for those, you'll most likely have the choice to stretch them, or just watch them as intended.