No, you do not need a garbage disposal to have a dishwasher. A dishwasher will work without one, and while it's rare to see a home with a dishwasher that doesn't have a disposal, it is possible. The two appliances pair well and are often installed together because a disposal simplifies meal cleanup.

However, your dishwasher should not be used as a replacement for a garbage disposal. Subjecting it to continuous food leftovers can lead to major clogs. If you choose not to install a garbage disposal, you should thoroughly rinse your plates or scrape excess food into the trash or a compost pile before loading them into the dishwasher to prevent plumbing issues.

The main issue with garbage disposals is clogging. This can mean a clogging of the pipes that expel the food along with water, or a clogging of the mechanical parts themselves. The problem lies in the solution because you can’t easily see clogs if they’re inside the system, and reaching in to dislodge stuck items is a risky proposal. Call a pro and ask what they would do, but if you can’t see the clog easily, and running the unit won’t free it, you need to let a professional handle the fix. 

Your garbage disposal is not draining because it might have a clog due to food waste, a foreign object, grease, oil, fat, or it may have a broken motor. When water doesn't flush out of the disposal, that's a key sign that your disposal has a clog or needs repair. 

These appliances certainly increase the number of things allowed down the drain, but they’re not foolproof. There are several items you shouldn’t put in your garbage disposal, according to manufacturers and technicians. Don’t put coffee grounds down the drain. The same goes for eggshells, potato peels, hard foods, onion skins, and any dry, expandable food. Most importantly, never put grease or oils down the drain.