I have a website with forms, controls, etc. User inputs should be sent to a C++ program running on the same PC as the webserver (busybox httpd). On the other hand results of the C++ program should be sent back and displayed on the website. I am a bit lost where to start and what to Google. Can someone point me into the right direction?
You need to create a HTTP server which will receive HTTP requests and send HTTP responses. The requests are sent by your browser whenever you enter a web page, send a form, etc...
In C++ you have different libraries that do this without having to do it by yourself which is an existing question on stackoverflow. I recommend CURL as I had good experiences with it, but other libraries must do well too.
If you have to entirely make the HTTP server yourself, you can look at the example of Wikipedia (they also provide explanation about the Hypertext Transfer Protocol).