The Big Players
Microsoft is heavily pushing its Bing Maps API. GPS company Garmin switched to Bing for the companion website to its tracking device in 2010. After a year of user revolt, Garmin now offers users a choice of maps.
Bing does not have customizable maps or StreetView, but it does have Birds Eye View, an angled aerial photo that gives a unique view of a neighborhood.
Web geography pioneer MapQuest has a number of APIs, including two that could be considered replacements for Google Maps. The flagship MapQuest API and its limitless cousin the MapQuest Open API, which is based on OpenStreetMap.
The Old Timers
DeCarta never quite gets the respect it deserves for its deCarta Maps API, which has been around since at least 2005. The company dared to charge for its service long before Google, with commercial pricing clearly available on its site. But it also lets you brand the map as you see fit and promises never to put ads on the map, even in the non-commercial version (which is free for twice as many views as Google).
CloudMade offered customizable maps long before Google styled its Maps. CloudMade was started by OpenStreetMap founders and has always been built off the map anyone can edit.
I like the Mapstraction API so much, I wrote a mapping book about it. With Mapstraction, you don't actually choose a mapping provider, at least not permanently. You can start with Google Maps and always have the option to switch to another provider by changing one line of code.
CloudMade created a new, OpenLayers-like Library from the ground up. You can use any tiles you wish, with CloudMade's available by default. The Leaflet API has become popular because it is a small, flexible library.
You can also use any of these options with your own tiles, which is what foursquare did recently using the MapBox API.
Google Will Remain Popular
Photo via Tyler Bell