Was Medieval Japan a monarchy, if not what were they?

There was an emperor, although he had little power. The country was run by the most powerful clan leader, called the Shogun. The Tokugawa clan was pre-eminent after about 1600, so their rule is called the Tokugawa Shogunate. The rule was not absolute, and there was constant bickering and warfare among the various clans. This situation continued until the Meiji Restoration in the 1860s and afterwards, which was a restoration of imperial power as well as the opening of Japan to the external world.