Laura Werlin, author of Cheese Essentials, offers these recommendations (in agreement with SKK and Mayo above):
The younger the cheese, such as mozzarella, the more important that it not have any mold.
Mold on semi-soft cheeses, such as Gouda and Havarti, won't hurt but won't taste good.
Mold is essential to soft-ripened cheeses, like Brie, which has a mold rind, but any mold that looks awful (!) probably means it's past its prime anyway.
Blue cheeses are all about mold.
Cheddar and other semi-hard cheeses can develop blue mold in the cracks. It won't hurt you but you can cut away from it.
Gruyere and other hard cheeses usually don't develop mold but when they do, as others suggest here, cut 1/4 inch away with a clean knife.

Trim the mold off, then taste a little. Sometimes the moldy taste goes deeper than that you can tell by looking.

Good advice from susan g. I also find wrapping cheese in wax paper, then in a paper towel and then putting it in a plastic bag or a plastic container extends its life quite a bit.

Do be careful to cut with a clean implement every time.

For hard cheese, just trim off the mold.

An old fashioned way to inhibit mold on cheese is to wrap it in cheesecloth (!) that has been dipped in vinegar, and wrung out.

This is what the Mayo Clinic says:
With hard and semisoft cheese, you can cut away the moldy part and eat the rest of the cheese. Keep the knife out of the mold itself so that it doesn't cross-contaminate other parts of the cheese. Cut off at least one inch around and below the moldy spot. With soft cheeses, however, the mold cannot be safely removed so they should be discarded. The same goes for any cheese that's shredded, crumbled or sliced.