They were an essentially apocalyptic sect. They associated the year 1666 with the number of the beast of Revelations, and believed that Jesus was about to begin the 1,000 year "fifth monarchy" (the first four were Babylon, Persia, Greece, and Rome.)
Two Fifth Monarchists were judges at Charles I's trial and signed his death warrant. After Oliver Cromwell dissolved the Rump Parliament, the Fifth Monarchists proposed that the country should be ruled by a body of "saints," based on the Old Testament Sanhedrin. These saints, they believed, would usher in the apocalypse and the second coming of Christ.
Oliver Cromwell actually tried to implement this scheme. English churches nominated 129 men for the new government; Scottish churches nominated five; and the Irish, six. Many of those nominated were Fifth Monarchists themselves or were otherwise sympathetic to their cause. This Parliament of Saints, (also known as the Barebones Parliament,) lasted a whole 6 months before it dissolved itself for the good of the people.
After that, Fifth Monarchists' power and influence waned. They tried and failed to overthrow Cromwell's Protectorate. They later tried to conquer London (on Jesus's behalf,) and many were subsequently killed hideously for treason.