1. What are the most common TV standards in the world?
NTSC: National Television Standards Committee PAL: Phase Alternate Line SECAM: SEquential Couleur Avec Memoire (Sequential colour with memory)
There are other differences though. Strictly they are just different colour systems, but most countries which use PAL have 625 lines in a picture and send 25 full pictures/second, most NTSC countries have 525 lines and send 30 full pictures/second (mostly for historical rather than technical reasons). That complicates things.
2. Who devised them, and when? and why? Are they as old as television?
The first serious TV experimenting was done in several countries around the period 1900-1930, mostly black & white. The BBC started a regular service in 1936, other countries followed soon after, but since the technology was developing very rapidly there were always improvements being made. The BBC started with 405 lines, the US started a service a couple of years later with 525, by the time other European countries started the technology allowed 625 lines. France even tried 819 lines.
All those system were black & white, but people wanted to have colour. During the 1940's much of Europe was at war, and technological development for entertainment slowed down, but in the US they were able to continue and devised a colour system which was compatible with the existing black & white one.
By compatible I mean that a black & white TV got a black & white picture, a colo(u)r one got a colour picture. No need to make people throw away their B&W TVs. This system was endorsed by the American National Television Standards Committee, and was named after it => NTSC.
After the war other countries started to look at colour. NTSC was a very clever system, but it had some flaws. Engineers in various countries tried to improve on it, and Telefunken in Germany came up with a simple modification which improved colour stability. It was named PAL because they reversed the Phase of the colour signal on Alternate Lines.
At the same time Henri de France, in France, fixed the same flaw in a different way. His design (SECAM) needed a memory inside the set which made it more expensive. PAL gave as good a result, so most countries opted for that. France stayed with SECAM, possibly because in the De Gaulle era of the 50's memories of German occupation were still fresh, and dropping a French system in favour of a German one would have been unpopular. Rumour has it that the French government subsidized Thomson to make memory affordable.
Since Britain went PAL, France went SECAM, and the US went NTSC, any colonies or dependencies of those countries tended to get the same system. India/Pakistan got PAL, Algeria got SECAM, and since the US helped rebuild Japan after WW2 it got NTSC, etc.
3. What's the difference?
To squeeze a colour signal into the same space as a black & white one, and stay compatible, the NTSC designers separated the colour and brightness information. The human eye is less sensitive to colour, so they were able to reduce the bandwidth of that signal (make it take up less space in each channel), 'hiding' it at the high-frequency end of the video. That meant they didn't need to make the channels bigger, and incompatible.
To do that, they used the fact that you can represent most colours with a combination of Red, Green and Blue. If you film a scene with three cameras, one for each colour, then add all the outputs together you get a black & white image. This signal is called luminance, usually represented by 'Y'. Mathematically Y = R + G + B. (Actually, not all the contributions are equal).
They then transmitted the Y signal just as for a black & white TV, and also transmitted the R and B in the extra colour signal. B&W TV's only saw Y, and colour TV's got Y, R and B. Since Y = R + B + G, G can be obtained as Y - (R+B), so they didn't need to transmit all three.
To get both R and B into one signal, they use a combination of Phase and Amplitude modulation (think of it as AM and FM at the same time). Its called quadrature modulation, and works very well, but is susceptible to phase changes as it passes along cables, etc. If the signal gets +10 degrees phase change the colour will visibly change, which is why NTSC TV's have a tint control.
PAL overcomes that by sending R and +B on one line, then R and -B on the next. That way a +10 phase shift on one line becomes -10 on the next, and small differences will cancel out. PAL TV's don't need tint controls. (Some old PAL sets may have a one, however).
SECAM doesn't send both R & B together, it sends R on one line, B on the next. No fancy modulation, so no phase problems, but you need a 'memory' in the set to save up the signal from the previous line, since both R & B are required together for processing.
4. Why do you need different TVs?
Mostly because of the different numbers of lines. Its quite easy to make one colour decoder which can cope with all the systems, but making a TV which can do 625 and 525 lines, 25 and 30 pictures/second, gets expensive. Consumers shop on price, no-one will buy a SECAM TV in the USA even if it only costs $20 more, since there aren't any SECAM channels.
5. Why do you need different VCRs? Why can't one VCR record the same "output"?
Some can, but like TVs it costs more to make them adjust. The motor speed varies with the number of pictures transmitted per second, for example. (This is covered in more detail in the document: Notes on the Troubleshooting and Repair of Video Cassette Recorders.)
6. Why did different systems evolve? Is one cheaper? Is one better?
When originally developed, expense was considered based on contemporary technology. As noted, politics may have been equally important.
As to which has better quality, its all rather subjective. The 625-line system adopted in Europe has better vertical resolution than the 525-line US system, but some people find the 50Hz field rate still produces some flicker. NTSC/PAL/SECAM are all equally capable of excellent colour reproduction, but under poor signal conditions NTSC can degrade more quickly.
7. Are there other systems besides the ones I've mentioned? Why?
Some others, like MAC where the colour and luminance are completely separated. That gets rid of interference (ever see the strange colours which appear on very fine check patterns?) but is more expensive and really only possible due to modern electronics.
8. Are there going to be more or less systems in the future?
That is THE question! There are certainly going to be different systems, more lines, better sound, etc.
9. Is there any way to convert a PAL tape to NTSC or vice versa?
Yes. If the PAL tape has 625 line pictures and the NTSC one has 525 line then you normally need a computer which can read in one format and re-adjust things. Not cheap, but becoming cheaper, several companies offer that sort of service. Some PAL VCRs can do a half-conversion, enough to fool most PAL TVs into thinking its got a PAL signal.
10. Do they teach this stuff in electrical engineering courses?
Sometimes. Some of it, depends a lot on the course and school.