What specific form of government in Western civilizations was unique?
The unique and most important form of Western government is Democracy, or Constitutional Republicanism, where the people are more important than the rulers. Before the rise of Athenian Democracy, almost all governments were monarchies, with one single ruler creating laws on the basis of royal blood or divine influence. With Democracy, the first examples of "power to the people" and parliamentary systems appeared. This existed in Athens and was copied in other places, but it wasn't until the Colonies in North America created the Constitutional Republic format of government that the concept became truly workable.
The change from a monarchist society allowed "the common man" to make more difference in their own lives, and to influence government to feel more useful. Electing officials to represent districts has existed for a long time, but in monarchist societies they had little actual power, since the monarchy could quash any rebellion through force of power. With Representative government, the government structure itself is, in theory, directly answerable to the citizenry it represents. This mode of government is more successful and more stable than most historical monarchies.