What's the purpose of gruntjs server task?

The server task is used to start a static server with the base path set as the web root.

Example: Serve ./web-root as http://localhost:8080/:

grunt.initConfig({
  server: {
    port: 8080,
    base: './web-root'
  }
});

It will function similar to an Apache server, serving up static files based on their path, but uses the http module via connect to set it up (source).

If you need it to serve more than just static files, then you'll want to consider defining a custom server task:

grunt.registerTask('server', 'Start a custom web server.', function() {
  grunt.log.writeln('Starting web server on port 1234.');
  require('./server.js').listen(1234);
});

And custom server instance:

// server.js
var http = require('http');
module.exports = http.createServer(function (req, res) {
    // ...
});

Can I use the server task mapping concatenated/minified files to test my application [...]

Concatenation and minification have their own dedicated tasks -- concat and min -- but could be used along with a server task to accomplish all 3.

Edit

If you want it to persist the server for a while (as well as grunt), you could define the task as asynchronous (with the server's 'close' event):

grunt.registerTask('server', 'Start a custom web server.', function() {
  var done = this.async();
  grunt.log.writeln('Starting web server on port 1234.');
  require('./server.js').listen(1234).on('close', done);
});

The server task is now the connect task and it's included in the grunt-contrib-connect package.

The connect task starts a connect web server.

Install this plugin with this command:

npm install grunt-contrib-connect --save-dev

Note: --save-dev includes the package in your devDependencies, see https://npmjs.org/doc/install.html

Once the plugin has been installed, it may be enabled inside your Gruntfile with this line of JavaScript:

grunt.loadNpmTasks('grunt-contrib-connect');

Run this task with the grunt connect command.

Note that this server only runs as long as grunt is running. Once grunt's tasks have completed, the web server stops. This behavior can be changed with the keepalive option, and can be enabled ad-hoc by running the task like grunt connect:targetname:keepalive. targetname is equal to "server" in the code sample below.

In this example, grunt connect (or more verbosely, grunt connect:server) will start a static web server at http://localhost:9001/, with its base path set to the www-root directory relative to the Gruntfile, and any tasks run afterwards will be able to access it.

// Project configuration.
grunt.initConfig({
  connect: {
    server: {
      options: {
        port: 9001,
        base: 'www-root'
      }
    }
  }
});