What are the most basic and common categories and sub-categories that you should have on a directory?
I wouldn't say that you should have anything in particular, often many look so the same. But for top il name a few, Shopping, Computers, Internet, Web Directories, Sports, News etc... For Sub Cats theres just too many to start suggesting. For ideas take a look at 5 directories that have different category structure, get some ideas, mix up what you find, and throw in some of your own. Look for a pattern of the categories you see in directories that never seem to get a listing, it may be worth leaving these out, theres quite a few that are never used, either in the dumps or copied from another directory when putting their own together. Im guessing that you were reffering to a general directory.
I've always used Yahoo directories as a reference point. Pick the most popular categories and mimic it.
Don't go for common, your only entering a market which is already saturated. Just a view but going with a niche category structure unique to you would prove the most beneficial.