What does double parentheses mean in a require

This is a pattern in which the module.exports of the module you are requiring is set to a function. Requiring that module returns a function, and the parentheses after the require evaluates the function with an argument.

In your example above, your ./routes/index.js file would look something like the following:

module.exports = function(app) {
  app.get('/', function(req, res) {

  });
  // ...
};

This pattern is often used to pass variables to modules, as can bee seen above with the app variable.

Well, require is a function provided by Node.js that basically loads a module for you and it returns whatever you expose in the module that you loaded.

If what you expose (through the use of module.exports) in a given module is a function, then that is what requires returns. For instance.

//moduleX.js
module.exports = function(){
  return "Helo World";
}

Then if you require it, you get a function back

var f = require('./moduleX');
console.log(f()); //hello world

Of course, you could invoke the function directly once you require it.

var greet = require('./moduleX')();
console.log(greet);

That mean that behind that, there is a function that is exported using module.exports:

module.exports = function(app) {
   app.get("/", function(req, res){});
}

See also http://www.choskim.me/understanding-module-exports-and-exports-in-node-js/

Sidenote:

You could create function on the fly:

A.js

module.exports = function(data) {
   return function(req, res, next) {
        // Do something with data
        next();

}

main.js

...
app.use(require("A")(data));
...