The two nicknames for the English Civil War?

Answer 1

There have been several civil wars in England, the most famous of which is simply referred to as the "English Civil War" where Charles I marched into parliament, looking for 5 Members of Parliament that had tried to commit treason, thereby causing civil war. (As the royal family are not allowed to enter the house of commons)

You may be thinking of the Baron's war, in which King John had refused to abide by the Magna Carta he had previously issued.

Perhaps, however, most famously, you are thinking of the War of the Roses, the fighting between the houses of Lancaster and York (whos symbols are respectively the red and white rose) for dominance of the British throne. The war was ended when Henry VII, or "Henry Tudor", a relatively unknown bloke from Lancaster house defeated Richard III, and married Elizabeth of York, to unite the two houses.

Answer 2

Roundheads given to the Parliamentary army and Cavaliers were supporters of the Monarchy