In Node, the __dirname is a global object that contains the name of the directory that the executing script resides from. For example, if you are running node script.js from /home/user/env, then __dirname will contain `/home/user/env.
The method app.use() is a function inherited from the Connect framework, which is the framework Express is written on. The method attaches a middleware function to the application stack, which runs every time Express receives a request.
The code you showed mounts a static server to the path / that reads from the directory the script is executing from:
app.use('/', express.static(__dirname));
If you were to change the path to /path, then the static file server will serve static files from that path instead. If you specify no path, then / is used by default.
As for what express.static() does itself, it accepts a path and returns a middleware function that listens on requests. This is how the middleware works:
- Check if the request method is 
GET or HEAD. If neither, ignore the request. 
- Parse the path and pause the request.
 
- Check if there is a default redirect. If so, redirect with a 
HTTP 303. 
- Define the handlers for if a directory or error is encountered.
 
- Pass everything to the send module for MIME identification and file serving.
 
Here is the static() middleware source from Connect for reference:
exports = module.exports = function(root, options) {
  options = options || {};
  // root required
  if (!root) throw new Error('static() root path required');
  // default redirect
  var redirect = false !== options.redirect;
  return function staticMiddleware(req, res, next) {
    if ('GET' != req.method && 'HEAD' != req.method) return next();
    var path = parse(req).pathname;
    var pause = utils.pause(req);
    function resume() {
      next();
      pause.resume();
    }
    function directory() {
      if (!redirect) return resume();
      var pathname = url.parse(req.originalUrl).pathname;
      res.statusCode = 303;
      res.setHeader('Location', pathname + '/');
      res.end('Redirecting to ' + utils.escape(pathname) + '/');
    }
    function error(err) {
      if (404 == err.status) return resume();
      next(err);
    }
    send(req, path)
      .maxage(options.maxAge || 0)
      .root(root)
      .index(options.index || 'index.html')
      .hidden(options.hidden)
      .on('error', error)
      .on('directory', directory)
      .pipe(res);
  };
};