Local governments take care of things like local roads, water, sewage treatment, schools, property laws. They enforce local laws like “you can’t step on the grass downtown” and stuff like that. There may be local police departments, they might have a sheriff who runs a jail. Anything your city or county government does.

If you’re looking for specific ideas get a phonebook and open it up to the blue pages. There will be listings in there for state and local governments. If you look at their departments you’ll get an idea of what they do. If you give specific examples of departments in state and local government where you live it will really impress your teacher, I bet. Good luck.