API = application programming interface
This is the 'library' of code bits that allows a developer to talk to a device, be it a kinect or netduino or arduino. A programmer like me will use C# .NET and write software that uses the 'library' (DLL) provided by the device manufacturer and the methods within it to talk or listen to the device.
One API will be provided with the Kinect.
Another API will come with the arduino - a.k.a. programmable controller board (PCB).
The .NET programmer will write software that will accept input from the Kinect (i.e. a guy waves his hand at the Kinect) and send that signal into a method within the arduino DLL. Sending the arduino an instruction (i.e. 'actuate servo') from the .NET program will complete your task.
1. buy the devices and install their APIs onto your development machine
2. download sample programs that operate Kinect and others samples that operate arduino
3. examine the sample programs for code that is similar to what your goal describes
4. write your own code to listen to the Kinect and speak to the arduino
5. you'll have setup the arduino to actuate your servo - and you don't ask about that part in this forum, but you can use Bing! and the arduino forums.
Good luck. Patience is required, as well as gumption and spunk. Deliver.