The Roundheads were a group of people who supported Parliament & Oliver Cromwell during the English Civil War. They were also called 'Parliamentarians'. They fought against Charles I and the Cavaliers otherwise known as 'Royalists'. The Parliamentary army put their soldiers in red coats. The British army continued using red in later centuries.
Some Puritans, but by no means all, wore their hair closely cropped round the head or flat and there was an obvious contrast between them and the men of fashion, who wore long ringlets. "Roundheads" appears to have been first used as a term of derision toward the end of 1641, when the debates in Parliament in the Clergy Act 1640 were causing riots at Westminster. According to John Rushworth the word was first used on 27 December 1641 by a disbanded officer named David Hide, who during a riot is reported to have drawn his sword and said he would "cut the throat of those round-headed dogs that bawled against bishops". A few decades later “Whigs” was the word for men of similar opinions.