Washington, D.C.-based startup MapBox offers a more customizable alternative to mainstays like Google Maps that primarily provide its users with professionally curated content and navigation. Its platform is fairly easy to use, but limited in its free (starter) version. It gets more functionality as its price tag rises.
Mapbox Studio is the free version. Beginners can start making simple maps very quickly.
For complicated maps, one should consider the cost of hosting it on the Mapbox server. Using the API might make more sense in that case, but it requires learning how to code with Javascript. Good for presentations for students and teachers. Below is an example from a map on NYC Citibikes in the MapBox Studio Gallery.