AngularJS : What is a factory?

I've been doing a lot of work on Angular.js and overall I find it to be an interesting and powerful framework.

I know there have been a lot of discussions on Services vs. Factories vs. Providers vs. Values, but I am still pretty confused about what a Factory is.

Factory has been defined in other StackOverflow discussions as the following:

Factories

Syntax: module.factory( 'factoryName', function ); Result: When declaring factoryName as an injectable argument you will be provided with the value that is returned by invoking the function reference passed to module.factory.

I find this explanation to be very difficult to grasp and it doesn't increase my understanding of what a factory is.

Would anyone have any explanations or real life examples to share about what exactly a Factory is and why you should use it in lieu of a Service, Provider, or other?

Update

A service holds a reference to any object.

A factory is a function which returns any object

A provider is a function which returns any function

-phew-

From what I understand they are all pretty much the same. The major differences are their complexities. Providers are configurable at runtime, factories are a little more robust, and services are the simplest form.

Check out this question AngularJS: Service vs provider vs factory

Also this gist may be helpful in understanding the subtle differences.

Source: https://groups.google.com/forum/#!topic/angular/hVrkvaHGOfc

jsFiddle: http://jsfiddle.net/pkozlowski_opensource/PxdSP/14/

author: Pawel Kozlowski

var myApp = angular.module('myApp', []);

//service style, probably the simplest one
myApp.service('helloWorldFromService', function() {
    this.sayHello = function() {
        return "Hello, World!";
    };
});

//factory style, more involved but more sophisticated
myApp.factory('helloWorldFromFactory', function() {
    return {
        sayHello: function() {
            return "Hello, World!";
        }
    };
});

//provider style, full blown, configurable version     
myApp.provider('helloWorld', function() {
    // In the provider function, you cannot inject any
    // service or factory. This can only be done at the
    // "$get" method.

    this.name = 'Default';

    this.$get = function() {
        var name = this.name;
        return {
            sayHello: function() {
                return "Hello, " + name + "!";
            }
        };
    };

    this.setName = function(name) {
        this.name = name;
    };
});

//hey, we can configure a provider!            
myApp.config(function(helloWorldProvider){
    helloWorldProvider.setName('World');
});


function MyCtrl($scope, helloWorld, helloWorldFromFactory, helloWorldFromService) {

    $scope.hellos = [
        helloWorld.sayHello(),
        helloWorldFromFactory.sayHello(),
        helloWorldFromService.sayHello()];
}​

One major difference I see is that you can run custom code in the factory. But, in a service, only object creation happens.

myJs.factory('Factory', function() {

    //Write custom code here

    return {
            Hello: function() {
            return "Hello, World!"
        }
    };
});