After the country adopted a new constitution that was largely imposed on them by American-led occupation forces, in 1947, there were two more events in subsequent years that were crucial in shaping the political structure of postwar Japan.
In 1952, the U.S. and most of its World War II allies entered into a peace treaty with Japan. One consequence was the end of the U.S.-led occupation of Japan. Before 1952, elections took place under the new constitution, the parliament met, and prime ministers were selected, but ultimate power rested with the occupiers, led by the American General Douglas MacArthur, until President Harry Truman fired him in 1951.
From 1952 onward, Japan had, theoretically at least, full sovereignty. But, in reality, that sovereignty is significantly limited by this clause in the 1947 constitution:
Aspiring sincerely to an international peace based on justice and order, the Japanese people forever renounce war as a sovereign right of the nation and the threat or use of force as a mean of settling international disputes. In order to accomplish the aim of the preceding paragraph, land, sea, and air forces, as well as other war potential, will never be maintained. The right of belligerency of the state will not be recognized.
This has been interpreted to allow Japan to maintain relatively small defensive armed forces. But Japan is largely under an American security umbrella. Some small countries, such as Monaco, for practical reasons rely on the military of a large neighbor (in Monaco's case France) for their defense. But I don't know of any otherwise-independent state anywhere near the size and wealth of Japan that has relinquished that degree of the right and duty of national defense.
The second big development was the 1955 merger of the Liberal and Democratic parties. That merger created what came to be perhaps the most successful party in any country that allows multi-party elections, the Liberal Democratic Party.
That name doesn't mean what most Americans in the postwar era probably think it means. The Liberal Democrats are right of center, with heavy support from rural constituencies and business interests. Not that the Democratic Party in the U.S. doesn't have ties to business interests, but it's not generally perceived that way.
Next: a history of elections and prime ministers.