You might investigate ODBC to see if there's a workable method for using that method.
Also look at whether your other application can import data and if so from what file formats. You may be able to export the data into a file from Filemaker and then import it into the other application.
An application programming interface (API) is a specification intended to be used as an interface by software
This is a good overview introduction to APIs and Filemaker
How to Integrate PayPal API's into FileMaker Solutions
December 9, 2011
Example of MailChimp API and Filemaker
This FileMaker Pro Tutorial is not an actual plugin or wrapper, but walks you through the steps necessary to get a basic integration working. No FileMaker plug-ins are required with the technique—a Web Viewer and a Custom Function do the heavy lifting to pass info back and forth. Check out Part 1 here and Part 2 here. The demo file is also available at the end of Part 2. Provided by Douglas Alder.