The only 'real' measure a digital photograph has of its resolution is its pixel dimensions. Dots per inch (dpi) is a measure of resolution only when printing, the number of 'dots' the printer makes for each inch of paper. The more dots per inch, the fewer inches the same size image will cover, so the image will be smaller on the page. But not on the screen. Technically, dots and pixels don't have to mean the same thing, and you can tell the printer to print out at lower resolutions (i.e. larger dots), but for most purposes, dots can be treated as printed pixels, so an image 3000 pixels in width, printed out at 300dpi will be printed at a width of 10 inches. 300dpi the most common for high quality prints, particularly when using consumer-level printers. Professional printers may be capable of much higher resolutions but even then 300dpi is often seen as adequate.

You can set the dpi resolution of a photograph using most photo-editing software, for example, Photoshop but also GIMP and others, but doing this has no effect on the digital image. The pixel dimensions remain the same, the image itself remains unchanged. The only place this has an effect is when you print it out, although even then, usually when you go to print a photo you have the option to set the print size yourself, overriding the dpi that you set. Basically, the dpi you can set in photo-editing software is only a suggestion that doesn't come into effect until you go to print.

If you're using the image in some kind of page layout software such as Illustrator or InDesign then most likely, when you import the image, it will be sized on the page according to the dpi that was set. However again, the dpi is only about the image's size on the page, and until you print it out you can still change that, making use of the full resolution of the original image. If you then start to resize the image to fit your layout then you're overriding that dpi and usually there will be a way to see what the actual dpi is.

In short, unless you're trying to streamline your workflow by having all your images automatically sized for a certain dpi, there is no benefit to setting the dpi within your photo-editing software. The dpi settings wherever you see them are just 'soft' settings that change the way something prints, not the underlying data.

The relevance of all this is that dpi is of course still important when you come to print the photograph, but I do sometimes get photographs sent back to me asking if I can change the dpi and it really isn't necessary for me to open the photo up in Photoshop, change the dpi, save it again and then send it back, and if you're working on a jpeg then all that will achieve is degrading the quality of the photograph because each time you re-save it you're also degrading it.