TypeScript Modules with classes in different files

As a general rule, never mix and match import/export with /// <reference> tags to files that contain anything other than declarations (i.e. only reference .d.ts files if you use import/export anywhere in your code). What you want is this (see NB additions/deletions):

/* file Router.ts */
///<reference path='../DefinitelyTyped/node/node.d.ts'/>
import Url = module("url");

// NB Removed containing module for clarity
export class Router {
    static route(url: string) {
        var pathname: string = Url.parse(url).pathname;
        var query: string = Url.parse(url).query;
        console.log("About to route to " + pathname);
    }
}

/* file Server.ts */
///<reference path='../DefinitelyTyped/node/node.d.ts'/>
// NB Removed reference

import Http = module("http");
import Router = module("./Router"); // NB Added

module Core {
    export class Server {

        private server: Http.Server;
        private port: number;

        constructor(port: number) {
            this.port = port;
            this.server = Http.createServer(this.onRequest);
            this.server.listen(this.port);
            console.log("Server started at http://127.0.0.1:" + this.port + "/.");
        }

        private onRequest(request: Http.ServerRequest, response: Http.ServerResponse) {
            var url: string = request.url;

            console.log("Received request: \"" + request.url + "\".");

             // NB first 'Router' is the import name at the top of this file, feel free to change
             // NB second'Router' is the name of the class from that module
            Router.Router.route(url);

            response.writeHead(200, { "Content-Type": "text/plain" });
            response.write(url);
            response.end();
        }

        public getPort(): number {
            return this.port;
        }
    }
}