OpenMRS has a Java API that provides access to the complex underlying data model. This API can and is used in many different applications, the primary of which is the reference "war" application.
Many languages are not able to take advantage of this api, however. For other external consumers of OpenMRS data and services we provide links through the web services modules.
REST Web Services
The preferred and supported method of web services in openmrs is through the REST Module. See that page for how to install and configure the module. You can also read about the REST Web Services API For Clients, and the provided REST Web Service Resources in OpenMRS 1.8.
(There is also an older Unsupported REST Module that provides simple access to patients, obs, and users. However, this will no longer be supported in favor of the previously mentioned module)