The emperor Go-Daigo was able to restore imperial power in Kyoto and to overthrow the Kamakura Bakufu in 1333. However, the revival of the old imperial offices under the Kemmu restoration (1334) did not last for long because the old administration system was out of date and practice, and incompetent officials failed gaining the support of the powerful landowners.
Ashikaga Takauji, once fighting for the emperor, now challenged the imperial court and succeeded in capturing Kyoto in 1336. Go-Daigo, consequently, fled to Yoshino in the South of Kyoto where he founded the Southern court. At the same time, another emperor was appointed in Kyoto. This was possible because of a succession dispute that had been going on between two lines of the imperial family since the death of emperor Go-Saga in 1272.
In 1338 Takauji appointed himself shogun and established his government in Kyoto. The Muromachi district where the government buildings were located from 1378 gave the government and the historical period their names.
Two imperial courts existed in Japan for over 50 years: the Southern and Northern courts. They fought many battles against each other. The Northern court usually was in a more advantageous position; nevertheless, the South succeeded in capturing Kyoto several times for short time periods resulting in the destruction of the capital on a regular basis. The Southern court finally gave in in 1392, and the country became emperor-wise reunited again.
During the era of Shogun Ashikaga Yoshimitsu (1368 - 1408), the Muromachi Bakufu was able to control the central provinces, but gradually lost its influence over outer regions. Yoshimitsu established good trade relations with Ming China. Domestic production also increased through improvements in agriculture and the consequences of a new inheritance system. These economic changes resulted in the development of markets, several kinds of towns and new social classes.
During the 15th and 16th centuries, the influence of the Ashikaga shoguns and the government in Kyoto declined to practically nothing. The political newcomers of the Muromachi period were members of land owning, military families (ji-samurai). By first cooperating and then surpassing provincial constables, a few of them achieved influence over whole provinces. Those new feudal lords were to be called daimyo. They exerted the actual control over the different parts of Japan, and continuously fought against each other for several decades during the complicated age of civil wars (Sengoku jidai). Some of the most powerful lords were the Takeda, Uesugi and Hojo in the East, and Ouchi, Mori, and Hosokawa in the West.
In 1542 the first Portuguese traders and Jesuit missionaries arrived in Kyushu, and introduced firearms and christianity to Japan. The Jesuit Francis Xavier undertook a mission to Kyoto in 1549-50. Despite Buddhist opposition, most of the Western warlords welcomed Christianity because they were keen in trade with overseas nations mainly for military reasons.
By the middle of the 16th century, several of the most powerful warlords were competing for control over the whole country. One of them was Oda Nobunaga. He made the first big steps towards unification of Japan by capturing Kyoto in 1568 and overthrowing the Muromachi bakufu in 1573.
Please read more about the rise of Nobunaga and the developments in the Azuchi-Momoyama period here.