Can anyone tell me anything about the last name Andersen? I'm looking for its origins but will appreciate any information that you can provide.

Andersen = son of Anders
Andersdatter = daughter of Anders

Norwegian family surnames were established by law in 1923. Before that most Norwegians used real patronymics, individual surnames. Some upper-class and town people had started using family surnames from about 1850.

In 1865 census there are about 27 000 men with given name = Anders. Their sons then would get Andersen as surname (individual). Later on these sons might start using Andersen as a family surname. So most Andersens are not related to each other.
Check the children of some 1865 census Anders given names: http://digitalarkivet.arkivverket.no/ft/sok/1865?fornavn=and...=

If you post the name of a real person from that Andersen family, with their birth and death dates and places, or anything you know, maybe we can work backwards from there and find out where YOUR Andersens come from.
Andersen could be Norwegian, could be Danish.