According to my dictionary, anamorphic is an adjective with the following meaning: Having, producing, or designating different optical magnification along mutually perpendicular radii.

Most of the DVDs I own state that the movie is shown in anamorphic widescreen. Those movies always look normal; I don't have to change the widescreen settings. Whenever a movie is not shown in anamorphic, I have to select a different widescreen setting in order to make the image look less distorted.

Back in the VHS days, the movies were modified to fit the screen, which means the anamorphic (or non-anamorphic) image had the sides chopped off. It appears there is more chopping going on.

I ordered the 20th Anniversary edition of Robocop and started to watch the uncut version. I noticed something different, so I popped in the older Criterion release of Robocop. It's not in anamorphic, so I had to adjust the widescreen setting. In the 20th Anniv. edition, they chopped off the top and bottom of the image, rather than the sides, to get it to fit the widescreen. They changed the image to anamorphic by chopping. I only noticed this because I have two versions, which allows me to compare.

My questions are:

Why do some movies get shot in non-anamorphic? Why do others get shot in anamorphic?
Why isn't everything converted to anamorphic? Why are sides and tops and bottoms sometimes chopped? Is it cheaper to chop than convert?
Why all the different aspect ratios? I have movies shot in 1.33:1, 1.37:1, 1.66:1, 1.78:1, 1.85:1, 2.35:1, 2.40:1. (As far as I know, all widescreens are 16x9, which equals 1.78:1. Most movies seem to be shot in 2.35:1, so, logically, widescreen TVs should be 21x9.)
Can someone explain to me what 'anamorphic' means? I'm not clear about the definition.

I can only tell you what I know which isn't an expert's view but anyway here goes - I don't know what the definition of 'anamorphic' is but what it amounts to is a squashed picture that gets unsquashed by the system it is played on. You see, most video storage mechanisms were designed to show a standard 4:3 ratio (I'm talking pal here but it's only slightly different for ntsc). So in pal, 4:3 is 720x576. 16:9 (widescreen) is 1024x576 but storage methods like VHS for example are still only capable of a 720x576 image. So the horizontal picture in widescreen would get squashed to fit within the 720x576 size. What you'd see is an image that looks tall and skinny. The widescreen television then outputs that at 1024x576, pulling out the horizontal image to fit resulting in a normal looking image (and, yes, reduced horizontal resolution).

As far as I know anamorphic DVDs are usually the same but I think it comes from tape storage mechanisms. So many things shot on tape are shot anamorphic to go widescreen. Not sure about film though. I imagine that is shot as is in widescreen and then converted to anamorphic during transfers.

The reason some aren't anamorphic, especially most VHSs and early DVD transfers is because on an old 4:3 tv, the image looks all tall and skinny because it's not being pulled out. So companies reckoned that not enough people had widescreen tvs and it would only confuse the 4:3 crowd. So they converted the 16:9 image to fit within the 4:3 frame leaving it letterboxed and meaning that anyone with a 16:9 tv would have to stretch it and it has a very low image resolution. US companies were particularly resitant to widescreen because they thought people with 4:3 tvs would feel like they weren't getting the full image they paid for. Of course, the opposite was true.

The problem was always that there was never a clean break from 4:3 to 16:9. For years most things that were shot for 16:9 would only be seen 4:3. Even now some tv shows (The Shield for example) were always 16:9 over here but 4:3 over in the US (even on the DVDs).

Hope some of that makes sense. I haven't answered all your questions but I'm not sure about some of the specific examples like Robocop. Though I do know on film there is often safe areas that were meant to be trimmed and on some bad transfers they are left in meaning the odd boom mic gets in when it wasn't supposed to. It's possible that the bits chopped in your recent Robocop version were never meant to be seen in the first place.

I assume you're from Europe, correct?

movies I buy on hd dvd or blu ray are mostly 2:35:1, tv shows in hd format are presented in 1:78:1 ratio.

1.78:1 is 16:9. If you multiply the 1.78 by 9 you'll see! It's just a different way of writing it. 2.35:1 is basically the film (35mm) equivalent of widescreen and is wider than 16:9 televisions. So when formatted to fit a 16:9 television, you'll get black bars at the top and the bottom unless it has been cropped (in which case it's incorrectly labelled, which happens).

Well, I turned to wikipedia, and there is a lot of information:

http://en.wikipedia.org/wiki/Aspect_ratio_%28image%29

As for the newest version of Robocop, the chopped material is actually part of the film. Rather than chopping the sides, they chopped the top and bottom. If not for the earlier Criterion release, I never would have noticed. Likewise, I didn't notice the chopped sides back in the VHS days because I had no idea they were doing it. I remember being resistant to watching widescreen movies at first, which is typical human behavior (resistance to change). Possibly, because I didn't have a cinema in my house, I didn't make the connection that movies on VHS were missing the sides. I did notice that widescreen VHS tapes had black bars and it disturbed me at first, but later I realized it is better.