DPI stands for dots per inch.  This term is interchangeable with PPI, which is pixels per inch.  When discussing digital scanning most companies use DPI because there is a universal acceptance of this term.  The simplest way to understand is that it is a measure of sharpness to the scanned image.  Theoretically the higher the DPI the better the image.  This is not necessarily true.  The higher the resolution, the more detail in the scan.  Some detail is a good thing but, some is not.  For example scratches and grain are not good.  Also higher resolutions create a larger file size, which can cause trouble for people who have hundreds or thousands of images.  

If you are having your images digitized a decision needs to be made.  Should you have them scanned at 300dpi or 600dpi?  The answer is a result of what your intention is once they are scanned.  If you intend to have prints made for yourself or family members then 600dpi is the best option for you.  It may be that your intention is to simply have a permanent record of your images digitally to view on your computer; if this is the case then 300dpi will suit your purpose.  Some customers chose to error on the safe side and have all images scanned at 600dpi just in case, other chose 300dpi because they already have print copies of their photos, and some use a combination of both.

Slide and negative scanning is much the same except the resolution is higher.  Winning Photography Solutions recommends 3000dpi for slides and negatives, any higher resolution is more trouble that it is worth and any lower will not give the quality our customers deserve.