The Mailgun API is built on HTTP. Their API is RESTful and it:
Mailgun has published Libraries for various languages. You may use our libraries, or your favorite HTTP/REST library available for your programming language, to make HTTP calls to Mailgun. Visit our Libraries page to see HTTP REST libraries we recommend.
To give you an idea of how to use the API, we have annotated our documentation with code samples written in several popular programming languages. Use the language selector at the top to switch between them.
For the full reference please check this site.