The Right Resolution
A good rule of thumb for scanning old photos is to set your scanner software so it'll create 300-dpi prints at whatever maximum print size you expect to make. If you have a computer capable of handling large file sizes, it is best to plan for a final print size of about 8 by 10 inches. For smaller photos, planning for a final print size of 4x6" or 5x7" is also fine.
If your scanner software makes you specify the scanning resolution of the initial image rather than the final print size, then you'll need to do some math. If you think you'll only want to reprint the photo at its original size or smaller, a resolution of 300 dpi should suffice. Otherwise, to get the correct scanning resolution, divide your target print size by the size of the original photo, then multiply that by 300. This print resolution reference chart from About Graphics Software provides resolutions for common print sizes. Keep in mind that a larger dpi or ppi translates into a larger file size. If your computer doesn't have a large amount of memory, you may not be able to handle scanning in at resolutions higher than 300dpi.
If you have the option, it is generally better to scan slides or negatives rather than prints. Photographs tend to fade and discolor faster with time, meaning that slides or negatives will generally provide a better image. You'll need to select a high resolution to create usable prints from slides and negatives, since their surface area is so small. For example, you'll need to set the scanner resolution as high as 3000 dpi to get a 8x10" print from a 1.5" slide.
After scanning your photo, be sure to save it with the lowest file compression possible. TIFF is the best format for this, but if you choose to use JPEG choose the highest-quality setting. Otherwise, you'll just be decreasing the quality of your image.
One last point. As you choose your scanning resolution and final print size, it is also important to take into account the quality of your original photo. If your original image is a small and grainy 4x6" photograph, you shouldn't try to create a final print that's larger than the original. Otherwise, you’re just magnifying the photograph's flaws.


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