As color encoding standards, NTSC was the first and simplest one. PAL solved a problem with over-the-air broadcasting of color TV programs of phase shift by inverting the phase of the color signal with every second line. SECAM, instead, used frequency modulation instead of phase shift modulation to encode color information, and alternated which component of color information was sent on each line (instead of sending both components in phase quadrature).
PAL and SECAM are typically used in Europe, where the basic video standard is 625 lines at 50 Hz, while NTSC is used in North America and Japan with 525 lines at 60 Hz.
As noted, Brazil makes use of the newer PAL method of color modulation while having the basic North American video standard.
But the terms "NTSC" and "PAL" are currently sometimes misused to refer to the 525 and 625 line video standards respectively, and I think that they're being referred to in this sense - because DVDs are referred to as being either NTSC DVDs or PAL DVDs.
The 525-line video standard runs at 30 fps, and the 625-line video standard runs at 25 fps.
DVD players are made to connect to TV sets, but they have an internal capability of doing framerate conversions, so you can (provided there is no region coding issue) watch a PAL DVD on an NTSC television and vice versa. (Brazilians, of course, would be watching NTSC DVDs natively, and PAL DVDs with conversion, despite using PAL as their color modulation system, since the framerate requires conversion - the color modulation system is trivial, the DVD player just has to provide the right signal.)
The thing is, of course, that most motion pictures with a soundtrack (silent ones used to be 16 fps, and that changed to 18 fps) are filmed at 24 fps.
So because the difference is very small, in Europe, when a movie is shown on TV, they just run the projector fast, showing the movie at 25 fps.
In North America, that isn't an option - in one sixth of a second, there are five frames at 30 fps, and four frames at 24 fps. This results in what was called "3/2 pulldown": TV is interlaced, so there are 60 fields per second - thus, if one takes the four frames of a movie that are in 1/6th of a second, and scans them this way:
Frame one: even field, odd field, even field
Frame two: odd field, even field
Frame three: odd field, even field, odd field
Frame four: even field, odd field
then, by including some redundant information in the video signal, one has the movie running at exactly the right speed.
I've always wondered why they didn't use a television standard with a 120 Hz field rate and quintuple interlace - but since the reason for using 60 Hz was so that power line effects wouldn't distort the picture vertically (so that TV sets could have cheap power supplies) a doubled frequency might not have done the trick.
Thus, one could have a 587-line video standard, with each field containing 115 2/5 lines. So there would be four equalization pulses added instead of one during the vertical blanking interval. This would divide up the extra resolution gained by eliminating the redundant fields when showing motion pictures evenly between vertical and horizontal.
