In Taiwan, the 1 January is Republic Day to commemorate the founding of the Republic of China in 1912 after the overthrow of the Qing Dynasty by the Wuchang Uprising. The holiday’s full name is “Founding of the Republic of China” day.
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It is important to distinguish Republic Day in Taiwan from National Day, which marks the outbreak of the Wuchang Uprising. Republic Day essentially marks its end, its success in the setting up of a provisional Republic in Nanking. Later, the capital city was moved north to Peking, however.
One might wonder why Taiwan has public holidays so tied up with events in mainland China. The reason is that the government of the original Chinese republic escaped to Taiwan in 1949 during the Chinese Civil War and continues there to this day.