There are problably a bazillion sites out there with good descriptions of how "widescreen" DVD material is displayed on different systems, but I will try and explain it simply. On a computer, the software determines that the disc is "widescreen" and will actually resize the window to 16x9, if it is played in a window. If it goes to full screen mode, well then it depends on whether your computer screen is 16x9 or not. If it is 16x9 then it plays full screen, if it isnt 16x9 then it usually letterboxes it.
For Televisions, it is actually a bit of chemistry between the disc format, the disc player and the user. If your tv is 16x9, then you need to setup you disc player (in the menus) to 16:9 tvs, it will then playback an anamorphic signal that your 16x9 tv will unstretch and play back full screen. If you have a 4:3 tv then your menu choices in the player are 4:3 letterbox or 4:3 pan&scan, and it will send out whichever signal you prefer.
This gets confusing as a producer, as you have just created an anamorphic dvd, but if some viewers have there silly dvd players setup to playback at 4:3 pan&scan, then your film will look like crap. And god forbid if they have it setup to playback 16:9, and they actually have a 4:3 tv. Then you will be looking at the squeezed anamorphic image.