The root API most applications use is the API built into the node software. It's as simple as starting a Docker container to get running and it doesn't take many resources.
Once you have a node you can use simple REST API calls to perform actions: RPC Protocol
There is a public API accessible here: Public bananode RPC API
It is only suggested for testing though and often runs out of database space
Known address API: BananoAddressAPI
More up-to date GoNano fork - https://github.com/gbl08ma/gonano