CUP has 3 APIs that can be useful to applications:
 
- SOAP API - Configuration/provisioning, presence status via polling or notification using XML over HTTP SOAP
- REST API - Configuration/provisioning, presence status via HTTP/REST
- SIP SIMPLE - Presence status, via SIP protocol
 
Not sure what you mean by 'modelled on which Operating System API?'

SOAP is XML over HTTP, which is not unix or windows (or device) dependent
Same with REST
SIP is a network wire protocol, so again not OS/device dependent
 
There are OS/language native libraries (available on the web, etc.) which can encapsulate the above - for example Apache Axis, which can generate native Java language code stubs for SOAP services (.NET has a SOAP compiler which does the same thing for Windows,) or a third party SIP stack which would encapsulate the SIP protocol in a C++ API.