'Patronymic' literally means 'father's name'. It means that someone calls himself after his father, for example a son of Jan would call himself 'Jansen'. This is similar to the English name 'Johnson'.
Patronymics were very popular before the introduction of the civil registration in 1811. Only when the civil registration was introduced, people were required to chose a fixed surname. Before that, people could call themselves however they want. In some regions, people named themselves after their fathers by using patronymics.
Even after the introduction of the civil registration, many people still used the patronymics. Especially in the early years of the civil registration, this was tolerated. The patronymic was used as a sort of middle name in addition to the official family name.
Patronymics were used instead of last names especially in the northern provinces of Friesland, Groningen and Drenthe and the northern parts of Overijssel.
In some southern regions like Noord-Brabant, patronymics were used in addition to last names. There, people often named themselves after more generations of ancestors. For example: A Willem Peter Adriaan Jan Verschuren would be Willem, son of Peter, son of Adriaan, son of Jan Verschuren. Beware however, because sometimes a son had the same name as the father and that name was skipped. Instead of Willem Peter Peter Adriaan Jan Verschuren, he might be called just Willem Peter Adriaan Jan Verschuren, which 'skips' a generation.