Grande format resolutions, as various folks have pointed out, depend on viewing distance. Several answers refer to line screens, but technology has moved on. Very few, if any, grande format jobs are printed that way, and in any case, none of the answers indicated how to translate from PPI (dots) resolution in Photoshop to LPI (lines of dots at different angles) resolution in offset. They are NOT the same thing. The related question e100 refers to has answers that also don't really cover the necessary bases.
There are two answers, depending on whether you have a images (raster) or shapes (vector, including text) or a mixture of both. As e100 pointed out, vectors are a Good Thing when you need to scale stuff up for grande format, but clients are clients, jobs are what they are, and we don't often get to choose.
RASTER (IMAGE) DATA
The overall, practical rule is: don't go much above the minimum resolution you need, to keep file sizes manageable and avoid problems with the printer's RIP (Raster Image Processor -- the software that translates you artwork to the physical dots that will hit the paper). Beyond a certain fairly definite point, which I'll cover, increasing the file size does nothing to increase output quality.
Let's start with offset, since most of us are familiar with it, then we'll look at big format stuff. Traditional offset printing uses four inks laid down in a definite order, "screened" using variable-sized dots arranged in lines spaced a certain distance apart. This is the origin of the term "line screen". The lines of dots are at a different angle for each color, carefully arranged to minimize the visual interference patterns called the "Moire effect." How far apart these lines are spaced gives us "lines per inch" (lpi). Newspapers typically use a coarse screen of 75 lpi, magazines 133 lpi, fine art magazines and books up to 200 lpi or more. This is NOT "dots per inch" (dpi).
You can figure out what minimum ppi you need for offset work by multiplying the lpi (given to you by the publication or print provider) by two to give you the ppi you should maintain in your image at full size. (There's good, sound math behind this calculation, involving esoteric stuff like Nyquist limits. -- I didn't invent it. I just work with it.) Thus for magazines, brochures and the like, 266 ppi (or dpi, depending on the program you're working with) is a good number if the final output is regular offset.
There is another, increasingly common type of "screening" for offset printing, that you'll see referred to as "stochastic" (which means "random") screening. This uses an irregular placement of the dots, giving more even gradations and finer detail for the same nominal "dot screen". For stochastic screen printing you can go as low as 200 ppi/dpi for handheld pieces, in practical terms. I've not seen any formal study on this, so this is partly empirical data from my own experience, proving out advice from printing industry sources.
300 ppi, which is often touted as "print resolution" is a generally-regarded-as-safe number that will work up to 150 lpi screen, but will not be enough for a high-quality 180 or 200 lpi job and is unnecessarily large for newsprint. It works well for that inkjet printer sitting beside your desk, however.
If you use a higher resolution than is necessary, the RIP will simply throw away the excess pixels and make its own decision about which ones to keep, so you don't gain by sending in artwork at 600 ppi for a 75 lpi job that only requires 150 ppi of image data.
Inkjet processes are very different from offset. They don't put pigment on the paper in 4 angled lines of dots, and the dots usually don't vary in size, so there's no such thing as "lpi" with inkjet. Because the dots are sprayed and overlapped, you can go as low as 150 ppi for about the same visual effect as 300 ppi/150 lpi in an offset printed piece. You can also push resolution as high as the native resolution of the specific printer (typically 600 to 1200 dpi, higher for professional desktop photo printers), especially if your image contains lots of very fine detail, but you don't gain anything by going above it. In practical terms, you can translate ppi in the artwork straight across to dpi on the substrate for all inkjet-based manufacturing methods.
For the sake of completeness, I'll throw in dye-sublimation printing, used in some photo printers. Dye-sub is about 300 dpi, far less than a typical inkjet. High-end dye-subs go as high as 325 dpi.
Grande format is almost invariably printed using an inkjet type technology, so these "straight across" ppi = dpi ratios are adequate and correct.
Grande format jobs are often scaled in Photoshop. A Lamar (national billboard advertising company) 48 foot by 18 foot billboard lays out in a 17.64 inch by 6.84 inch Photoshop file at 300 ppi according to their published spec, which includes generous bleed on all four edges. When you do the math, that's 9 dpi "in the air", which is pretty typical for that market. 6 to 12 dpi is the normal range. You have to scale billboards, obviously, because you can't input 48 feet as an image width. But you would also have to scale a 3m x 5m image if you want to output as a PDF (which you do -- I'll get to that under "Output").
Illuminated advertising, such as the pedestals you see in your local mall, are at a high resolution of 150 ppi because they might be looked at from 3 feet/1 meter, but they can easily go as low as 75 ppi, especially if they don't have a lot of fine detail, without problems.
Vinyl banners of 10ft/3m or greater never need to be more than 50 ppi. (Finally! The answer to the original question!) The intended viewing distance is the deciding factor, and when a piece gets to be 10 ft wide, 10 feet away is about as close as you can get for comfortable viewing.
Without getting unnecessarily mathematical, anything that will be see from 10 feet away (across a small room) should be at most 75 ppi at full scale; at 20 feet or more, 30 ppi is plenty.
All that said, there is a BIG caveat about images: don't up-rez an image in Photoshop and think you have a "higher resolution image." You don't. What you have is a bigger fuzzier approximation of your original image with the same amount of image information padded out by what the software guessed the extra pixels ought to look like. An original, unscaled image printed at 50 ppi will look at least as good as, and usually better than the same image pseudo-enlarged in Photoshop to make it "150 ppi". And never, never, never take a super-compressed jpeg that your client "got from the website" and try to scale it up for print. Those jpeg compression artifacts get uglier and uglier the more they're scaled up.
VECTORS and TEXT
Shapes and text should be created and output as vectors. Because a vector image is composed of mathematical expressions, not pixels, it scales to any size. The trick is to keep that vector information in your output file and not rasterize it too soon, especially if the job is going to end up on an offset press.
When text or vectors pass through the printer's RIP to go to a printing plate, they are automatically turned into a raster format, typically at 2400 to 2800 dpi. Examine a printed picture and its caption in any issue of a magazine, and you'll see at once that the edges of the text are much more precise than the edges of objects in the image. For this reason, advice to output text at 300, 600 or even 1200 dpi is misplaced. You will only degrade the quality of the final output by rasterizing text or vectors before you send them to press.
When you send work to an inkjet press, vectors end up at whatever the output resolution is of the particular machine. For grande format, that's a top end of 1200 dpi, more often 600 or less, depending on the size of the output, so preserving vector information doesn't gain you quite as much, but it's still better to put that conversion off to the last possible moment rather than doing it yourself.
A jpeg, tiff or png is a raster image. That's fine if it's all a photographic composition. But if your file includes text or vectors (such as a company logo), the fine resolution of the vector information is lost. Avoid this if possible. "Well," you say. "I'll just keep it as a native Photoshop file." Unfortunately, that often doesn't work out, because what most programs (except Photoshop itself) extract from a PSD is the flattened tiff copy of the composite image that is saved within the PSD, not the native Photoshop information. (Worse, unless you included the actual fonts that you used in the project, the printer's prepress department may use your native Photoshop file, but substitute a similar-but-not-quite-the-same font that will alter your design. The chances of that coming out better than you intended are... well, images come to mind of snowballs trying to survive in furnaces.)
The answer is to output a PDF from Photoshop, which preserves all the vector information as vectors, while leaving the raster data intact.
For best results (as they say on the label) TALK TO THE PRINT PROVIDER to find out what flavor of PDF they prefer, and if possible have them send you their PDF spec (often in a file with an extension of .joboptions) for you to follow. If the provider can't handle a PDF, it's a good sign you need to shop around for a different print shop or some serious lucky charms.