The flag of Taiwan is known as the "Blue Sky, White Sun, and a Wholly Red Earth." It belongs to the Republic of China, and its continued use is bound up with the question of what the ROC now is.
From secret society emblem to national flag
Lu Hao-tung, a revolutionary scholar, designed the sun emblem in 1895, and Sun Yat-sen set it on a red field in 1906 to produce the flag as it looks now. When the Republic of China was founded in 1912 it took the five-colored flag as its national flag and used this one as a naval ensign. The Republic of China adopted it as the national flag on December 17, 1928, after the Northern Expedition brought the country under Nationalist government.
The ROC carried it through the civil war against the Communists. When the war ended in 1949, the People's Republic was declared on the mainland and the ROC government withdrew to Taiwan, taking the flag with it. It has flown there ever since, one of the few things about the ROC that did not change in 1949.
Sun, sky and earth
Blue stands for liberty, white for democracy and red for fraternity, a set that maps onto Sun Yat-sen's Three Principles of the People: nationalism, democracy and the livelihood of the people. The sun's twelve rays represent the twelve months of the year and, in the traditional Chinese reading, the rightness of basic principles and of justice.
Where it can and cannot be flown
In Taiwan the flag hangs on government buildings, schools and military installations, and official protocol governs how it is displayed and handled. Abroad it is a different matter. Taiwan's international status restricts where the flag can appear, though it has been shown at events such as the Olympic Games under particular arrangements. Domestic variants exist for the military and for maritime use, keeping the core design and adding symbols for the specific service.
The flag is central to National Day on October 10 and to public events through the year. At international events its appearance sometimes causes a diplomatic problem, which is part of why supporters of Taiwanese sovereignty are so attached to it. Civic groups and political movements use it as a marker of identity as much as of government.
What it shares with the PRC flag, and what it does not
The white sun connects the flag to earlier Chinese revolutionary designs, and its blue, white and red carry the ideals of those movements. The People's Republic flag went a different way, a red field with stars for Communist principles. Taiwan's flag kept the pre-Communist revolutionary imagery, which is the clearest visual statement of the split.
References
Government of Taiwan Official Site
International Vexillological Association Publications
Historical Records from the Republic of China Archives