Athenians had a clear understanding of the difference between sovereign power and executive government, and were careful to keep the two separate. More careful than all modern democracies in fact. The US President not only heads the executive but has influence on what laws are passed. In the Westminster system the Prime Minister is the leader of whichever party holds the most seats. In both cases the sovereign power and executive duties are mixed. An Athenian would have called that sloppy.
By classical times there were nine archons: the Eponymous Archon, the Polemarch, the Basileus, and six others who served as magistrates.
The archons were elected for a year and after serving, could never serve again. If you're wondering how Pericles managed to stay influential for so long, it's because he was at no time an archon! Seriously. He got himself elected year after year to the part of executive government for which there was no limitation: Pericles was an almost perpetual strategos - a military commander - equivalent to being a member of the modern Joint Chiefs of Staff.
There were only three archons originally in archaic times: the three with the über-cool titles. The six magistrates were added as the population grew and the workload became too much.
The Eponymous Archon was in charge of the affairs of citizens. He was something like a city mayor. The Eponymous Archon was the one after whom the year was named. The Pericles Commission takes place in the Year of Conon.
The Polemarch was in charge of the affairs of the many resident aliens, called metics. In archaic times the Polemarch had been the war archon. The word Polemarch is conjoined of war and leader. Military command later became too big for one man and passed to the strategoi. The Polemarch then became in effect the equivalent of the Eponymous Archon for the metics.
The Basileus was the archon in charge of religious and artistic festivals. Basileus means King.
It's obvious if you look at their combined duties that the three main archons were a replacement for the ancient king of Athens. The Eponymous Archon managed civil affairs, The Polemarch led the Athenians at war, and the Basileus led them in worship. No one's too sure when or how it happened, but at some point the kingship was replaced with this triumvirate.
There were a few years in which social upheaval prevented the election of archons. Those years were considered ones of an-archy, literally no-archon. Our word anarchy comes to us direct from ancient Athenian politics.