If you want to keep a heavily-planted, lightly-stocked aquarium you can get away without a filter if you pick the right fish. You would also need to feed live food like Daphnia or Tubifex worms that will stay alive until eaten. You would still need to do water changes and you still need to provide water movement. It takes a lot of knowledge to know how to set up one of these systems. Most beginners would be disappointed with the options.
Filters allow us to keep more fish and have a more active aquarium than would be otherwise possible. They allow us to feed prepared foods that are more convenient and provide a balanced diet to a variety of fish. Filters keep the water clear by removing suspended particles. Filters build up colonies of bacteria that detoxify fish waste into less toxic forms, which still need water changes to completely remove. Filters can also hold activated carbon which removes compounds that can give the water a tint. Filters also provide water movement and help oxygen and carbon dioxide exchange in the water.
Filters are not strictly necessary for all fish and all aquaria, but they give even those tanks an extra margin of safety. They also allow us to keep enough fish for a lively and colorful aquarium. They also reduce the maintenance so that many people can enjoy having an aquarium without it becoming too much of a chore.