The term “Roundheads” was used to refer to one of the sides in the English Civil War that lasted from 1642 until 1651. This was not an official term for the side and they did not like the term. However, it has become the standard way to refer to this side. The Roundheads were the side that was in favor of more Parliamentary power and less power for the monarchy.
The English Civil War was fought between those who wanted more power for the monarchy and those who wanted more power for Parliament. The latter group, called the Roundheads by their enemies, generally wanted to have a constitutional monarchy of the sort that eventually came about after the Glorious Revolution of 1688. The other side was more in favor of a relatively absolute monarchy.
There were also some religious differences between the two sides, which is relevant to this question because it helps to account for the term “Roundhead.” The Parliamentary side was supported by essentially all the Puritans in England. The Puritan men generally wore their hair very short at a time when much longer hair was the fashion. This made their heads look rounder than those of men with long hair. For this reason, they came to be called the Roundheads.
Parliamentarians they wanted more power for parliament and less power for monarchy