You could use the HTTP Basic authentication scheme, which uses the Authorization
HTTP header. With Basic authentication, the client must provide its credentials on each request. so you might prefer the Digest authentication scheme, which is a little more secure.
Without more details, I might recommend OAuth 2.0 with the Client Credentials grant type. Basically, the client uses Basic authentication with its client credentials once to receive an access token, and then it uses the access token on subsequent resource requests. Typically the access token is submitted using the Authorization
HTTP header.
OAuth is really an authorization framework, but it provides a good solution for API authentication also, and you might find its authorization tools useful for securing access to API resources. It could be that the Authorization Code grant type is applicable to your use case as well.
Related: Best Practices for securing a REST API / web service