Serial Echo code breakdown [beginner]

There are a few geeky programming things in that sketch. It's normal if not everything is clear.

The 'unsigned' for MAX_INPUT is not needed, it can be a normal integer.

This is a declaration of a function:

void process_data (const char * data)

You can read it as this:

something_returned NameFunction (parameters)

The "void" in front is what is returned. In this case nothing, and the "void" word is used instead of "nothing".
The "process_data" is the function name, it can be anything you made up.
The parameters is in this case "const char * data". If you see a '*' you should say 'pointer'.
It is: "constant character pointer data". Meaning that "data" is pointer to characters.
The "const" is just added to indicate the the data is not changed inside the function.