When a dishwasher fails to completely empty the water after use, what's the most likely cause? Is its drain clogged or is it more likely a problem with the pump getting old and weak? If it's clogged, what's the best way to fix?

Chances are that the pump impeller is shot, in some models you can replace the impeller, in others you have to replace the whole pump assembly.

If your dishwasher drain tube connects to a garbage disposal unit, then you may want to make sure the disposal is clear. Also, if the dishwasher is draining but leaving some water, check for a kink in the drainage hose from the dishwasher to sink. I had the same problem and fixed it by straightening the hose.