A factory creates objects of a certain type. The factory takes an ID of some sort as an argument and creates the object which has been registered with the factory with the respective ID.
This code sample shows one way to make a factory. All objects are stored in an interal (intrusive) list inside the factory together with their ID. To create an object given its ID, the factory searches the list for the ID and calls the copy constructor of the object to create a new object of the same type.
A singleton is a class which controls the number of objects to be created from this class. Most often there shall be just a single object from a certain type. The basic principle to reach this is to limit the access to the constructor, i.e. making it private. There is a static function which allowes a single point of access to the single object.
Controversial discussion There is a controversial discussion if a singleton is good design.