Bernhard,
There are several ways to do this, and it really depends on what you actually need to do. A screen pop? Easiest method is to use Avaya IP Agent or One-X Agent application and have it open an application or URL and pass ANI in the command line. Otherwise, I need to know more to advise better.
EDIT:
For IP Office switches you can use TAPI interface provided by TAPILink software that is bundled with IP Office. For Communication Manager (Aura, Definity) you will need Application Enablement Services (AES) server that provides several APIs. For C# I would suggest either TSAPI or DMCC.
You can find tons of information and docs on http://devconnect.avaya.com. You will need to register to get access but it's free.
Regards,
Alex.