My experience is that many Americans have problems understanding the naming conventions used earlier in Denmark, especially those among the peasantry and worker families. Usually you can not follow a family name in these cases.

 
There are four categories of names: patronymic (named from the father), occupational, nickname or place name.

In the old days Danes were identified by their Christian name and their father's name: Hans Pedersen (or Pederson, Pedersøn), i.e. Hans, the son of Peder; or Maren Pedersdatter (or Pedersdotter, Pedersdtr.), Maren, the daughter of Peder.

Through the years I have seen some examples of family histories where the genealogist has not been aware of these facts. Lots of effort has been put into a search for the wrong person. If one of your ancestor's name was Hans Pedersen he cannot be a son of Ole Pedersen (Pedersen is not a family name, but a patronymic), but he can be a son of Peder Olsen, or another man called Peder. You can rule out all persons with a Christian name other than Peder.

When a man and a woman married, the woman did not take the name of her husband as is common today. Both of them kept their forename and patronymic as before.

There are of course some exceptions from this rule. Upper class families and many families in the cities usually used a family name. This family name could be the same as a patronymic, thus making it difficult to distinguish between a patronymic and a family name when we are dealing with families in the cities. You can also find some examples of farmer families that used a family name, but there are not many of them.

Family name derived from occupations, it was then easy to distinguish Hans the baker from Hans the hunter. Here are some names in Danish  with a translation to English:
Jæger meaning hunter, Bager meaning baker, Skrædder meaning tailor, Pagter meaning renter, Skriver meaning writer, etc.

The word "nickname" is derived from "an eke name," or added name. In the sense all surname began as extra names, so technically speaking, all surnames are nicknames of one sort of another. In our classification scheme, hereditary family names based on nicknames often describe an ancestor's appearance (stature, hair, eyes, complexion, size), a characteristic or trait (strong, bold, brave), financial status, habits or special skills. Here are some example: Lille meaning small, Rød meaning red, read, rede, reed, bay, gough, rudd, ruddy, ruff, russ, russell, rust. Grøn meaning green, hunter, an inexperienced. etc.

Surname sprang from place names in several ways: when someone was associated with or living near or by a particular hill, brook, bush, dale, valley, island, bridge, meadow, road, village or farm; Here are some example: (gaard, rud) in the end of a surname means it is a name on a farm, (bjerg, høj, bakke) in the end of a surname means someone lived close to a hill or mountain. etc.

A complicating factor when we are talking about Danes that emigrated to America is what they called themselves after they had settled down there. Most of the emigrants changed their name somehow. Danish names are usually difficult to pronounce in English and a change was very often the only solution. The Christian name usually was somehow Americanized, while the last name could in some cases be changed completely. One example I have is a man from Hejls with the last name of Mainz. In America he changed his name to Mines. That have been some problems for the American Decentness to track the name down in Denmark.
Very often they took their patronymic as their American family name. But we have many examples also that they took their father's patronymic as a family name. This fact makes it even more important that there is enough information about emigrated Danes to be able to identify them.

Today all Danish use family names. Patronymics are not used any more. The transition started in the late 18th century, and in 1904 the use of family names was made compulsory by law. Many family names today are originally patronymics that became family names. Practically all Danish family names today ending with -sen are originally patronymics.

The 10 most conmen family names in Denmark