The Final Destinations
The vast majority of images created or prepared in Photoshop and Illustrator are destined for the World Wide Web or for print. The difference between the two is substantial, and you must consider the ultimate use of your image when selecting a file format.
Check the documentation for the program in which you'll be using the Photoshop artwork. You'll want to check for information on file formats that can be opened or placed in the program, color mode and bit depth requirements, limits on file sizes, applicability of alpha channels or clipping paths for transparency, and any other factors that could determine which file format is most appropriate.
Web Versus Print
The World Wide Web uses certain standards in an attempt to ensure that everybody can see everything in a Web page, as long as the viewer has a decent computer and a relatively recent Web browser. These standards include certain types of file formats for graphics so that all browsers can display images from the Web.
Page layout programs accept certain file formats that allow for the best reproduction of the image with ink on paper. Illustration programs that you use with Photoshop are able to create graphics for both print and the Web.
Because the requirements are different, and even the color mode differs between the Web (RGB) and print (CMYK), the file format you choose should depend on the image's final destination.
Generally speaking, the print file formats are TIFF, EPS, and PDF. For Web, Photoshop/ImageReady and Illustrator can produce GIF, JPEG, PNG, and WBMP files. Illustrator can also generate SVG and SWF files for the Web. In addition, a variety of other file formats are available for different purposes.
Commercial Printing Versus Inkjet Printers
Keep in mind that there's a huge difference between preparing an image for commercial printing and preparing an image for output to an inkjet printer. With the exception of those using a hardware or software RIP (raster image processor), inkjet printers require RGB color data. The overwhelming majority of inkjet printers are used with programs that do not support CMYK color. (CMYK color mode is available only in professional-grade graphics and layout programs.)
Another important difference is that images prepared in Photoshop that eventually are printed commercially are usually placed into a page layout program. Inkjet-destined images are typically printed directly from Photoshop. Additionally, inkjet printers can output virtually any file format created from Photoshop.