This depends on the source and the intended purpose;
For web use, the format is less important - by the time the image is scaled down and compressed enough for viewing (jpg, or PNG normally), the artifacts of the original format and compression are often no longer visible. Web images are displayed at 72dpi in RGB format, and must be small enough to load quickly on a "typical" user's system. Typical user's vary by application, target market and country.- In Canada broadband access is prevalent, but in the USA a much larger percentage of people are still on dialup.
For print use, the format is important. The best format often depends on the source. If your files were produced in a vector format, such as Adobe Illustrator (.ai), Corel Draw (.cdr), or other vector program producing postscript (.eps) then you may already be in the best format. Remember to embed all fonts or convert them to outlines or paths (as applicable in your application). Also, any raster elements must be saved to print at least 300dpi at full scale.
Raster files should be saved in either PC or Mac formatted TIFF (.tif, .tiff), PhotoShop (.psd), or Portable Network Graphic (.png). Jpeg files can be used (and are often the native format for many scan programs and digital cameras) but compression ratio or quality setting is key here. With all raster formats the use of any "lossy" compression ratios can reduce print quality - when uncompressed for printing, these formats will show artifacts of the compression and thereby reduce the quality of the print. Use the highest quality setting if you are uncertain. Remember that the file must be big enough to print at at least 300dpi at your intended print size. For example, a 16" x 20" print would require a file at least 4800 pixels x 6000 pixels. Larger prints are possible if the intended viewing distance is increased, but up close a larger print may look grainy.
Also remember to convert your files to CMYK mode for printing.