Open the dishwasher door with a flashlight in hand. This is the whole inspection. You are looking for what the machine asks of its owner, which is less than most people expect and more than many people do.
The filter
Most dishwashers have a removable filter at the bottom of the tub. Lift it out and rinse off the food and grit under warm water. Check the screen for a film or crust that the rinse did not clear. A clogged filter leaves residue on glassware and can keep the drain from emptying fully. Set it back until it seats flat or clicks into place, depending on the model.
The spray arms
Turn each spray arm by hand. It should spin freely without catching. Look at the nozzle holes at the ends of the arms. If your model lets you remove the arms, clear any hard water deposit or debris with a thin wire or a toothpick. Refit each arm on its hub before you close the door.
The door gasket
Run a finger around the rubber seal. Look for tears, flattened sections, and grease. Wipe it clean. A gasket that no longer seals lets steam escape and lets water sit in the channel. Replace the gasket when it cracks or tears, not simply because it looks dull.
Hard water and the rinse aid
Where the water is hard, expect a white film on glassware and a deposit on the inner walls. Find out how hard your water is from the water utility or a home test kit. Then set the detergent and rinse aid settings to match what the owner's manual recommends for that level.
How often
In a busy household, check the filter and spray arms every few weeks. Check the gasket every few months. Run an empty hot cycle with a dishwasher cleaner when the glassware starts to look cloudy or the interior smells sour. Keep the owner's manual with your house file so the care steps stay in one place.
When it is asking for service
If water stays in the bottom of the tub after a cycle, or the machine stops with an error code, check the filter and the drain hose first. Clear what you find, then run a short cycle. If the problem remains, call a service person and give them the model number and the install year. The dishwasher page on this site shows typical service life and when to plan for a replacement.