DPI = dots per inch is only relevant when you output digital images to an output device, such as a printer (or a display,) since DPI is a measure of how many of pixels the output device will show on a line that is one inch long.
That means that the DPI measurement is totally useless as long as you aren’t printing. Video and things like DVD menus have a fixed size in pixels (called resolution) and DPI has no relevance. As long as you create the document with the correct size in pixels, you can set the DPI to 1 or 3,000, it doesn’t make any difference.
To demonstrate this, open Photoshop with any picture and choose the command “Image Size…” from the “Image” menu and uncheck “Resample Image”. You can then easily see that if you change the “Resolution” in pixels/inch, the actual pixel dimensions aren’t altered, only the printed width and height are changed!
If you want to measure the DPI of a DVD menu, you must play the DVD and measure with a ruler on the TV set itself. You’ll get a much higher DPI resolution on a small portable player than on a big hunkin’ 60-inch plasma display, since you you are pushing the same amount of pixels to displays of different sizes.
The pixel resolution in square pixels (which is the preferred format when working in for example Photoshop) for NTSC Widescreen is 864×486 pixels, for NTSC 4:3 it’s 720×540, for PAL Widescreen 1,024×576 and for PAL 4:3 it’s 768×576. Don’t forget the action-safe and title-safe either!
PS: The correct term is actually PPI (pixels/inch) for displays. The term “dots” comes from the raster dots that are traditionally used in printing presses.
PS. The only time the DPI/PPI is relevant is when importing/exporting graphics, like this Flash bug.