A typical traditional Danish name model in the old days would look like this:

Father: Anders Hansen
Mother: Ellen Magrethe Pedersdatter
Son: Christian Peter Andersen
Daughter: Johanne Cathrine Andersdatter
In other words, the last name of each generation depends on the first name of the father of the children.
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With the “name law” in 1828 it was decided to have unchangenable family names in the royal kingdom. After this year it was not common to use patronymic anymore (name derived from that of the father or a paternal ancestor usually by the addition of an affix ). The last examples of  patronymic naming in Denmark can be found in the north west of Jutland around 1882.

The law of 1828 also decided that patronyms that were already in use could be used as a future family name BUT it was a demand that the ending had to be -sen.  This was the origin to the many Danish sen-names like Jensen and Andersen.

Patronymic names are still used in Iceland and on the Faraoe Islands.
A new law Danish law from 2006 is again opening the posibility to use patronymic  last names. According to the law a patronymc can be made from either the mother or the father’s first name following either -sen eller -datter.