The question is valid, but way too vague to be answered with any specificity. In general, the resolution you need your image to be depends on:
- the type of image
- the size of the final piece
- the type printing technology being used
- what it's being printed on
The one specific answer is web graphics: there is no concept of DPI for images that will be on a screen. You only need to deal with pixel dimensions.
When printing, you have to deal with things like the type of printing (screen printing would be lower resolution than offset lithography, for instance), the type of substrate (printing on clay coated paper would typically use a much higher DPI image than printing on newsprint would), the type of image (images printed with a line screen [such as photos] don't need to have as high of a resolution as line art would need), the size of the final piece (a billboard doesn't need the same resolution as a glossy brochure), etc.
BROADLY speaking, photographs are typically set at 300DPI. Line art somewhere between 600 and 1200DPI. Large format printing is typically 100DPI or thereabouts.
Note that the terms dpi and ppi are sometimes used interchangeably. TECHNICALLY, we should usually be using PPI when it comes to image resolutions for printing but DPI has been a long used term as well. Search for DPI and PPI on this site to see lots of discussions on that topic.