The currently preferred (Exchange 2013 and 2016) API is EWS. It is purely HTTP based and can be accessed from any language, but there are .Net and Java specific libraries.
You can use EWSEditor to play with the API.
Extended MAPI. This is the native API used by Outlook. It ends up using the MSEMS
Exchange MAPI provider, which can talk to Exchange using RPC (Exchange 2013 no longer supports it) or RPC-over-HTTP (Exchange 2007 or newer) or MAPI-over-HTTP (Exchange 2013 and newer).
The API itself can only be accessed from unmanaged C++ or Delphi. You can also use Redemption (any language) - its RDO family of objects is an Extended MAPI wrapper. To use Extended MAPI, you need to install either Outlook or the standalone (Exchange) version of MAPI (on extended support, and it does not support Unicode PST and MSG files and cannot access Exchange 2016). Extended MAPI can be used in a service.
You can play with the API using OutlookSpy or MFCMAPI.
Outlook Object Model - not Exchange specific, but it allows access to all data available in Outlook on the machine where the code runs. Cannot be used in a service.
Exchange Active Sync. Microsoft no longer invests any significant resources into this protocol.
Outlook used to install CDO 1.21 library (it wraps Extended MAPI), but it had been deprecated by Microsoft and no longer receives any updates.
There used to be a third-party .Net MAPI wrapper called MAPI33, but it is no longer being developed or supported.
WebDAV - deprecated.
Collaborative Data Objects for Exchange (CDOEX) - deprecated.
Exchange OLE DB Provider (EXOLEDB) - deprecated.