Taiwan (officially named the Republic of China) is an island nation that is widely considered an independent nation, with its own democratic government. Mainland China, however, still claims it as its own territory.
Taiwan was originally part of the government of mainland China. The Republic of China was formed in 1912 to oppose both mainland Chinese rule, as well as Japanese. The Japanese ruled it in the late 19th century, with that rule ending at the end of World War II, and thus the Republic of China was recognized as the formal government. After the Chinese civil war, the losing Nationalists formally solidified control over the island, using military force to keep any suspected Communists at bay.
In the mid 1980s, Taiwan began to convert to a more democratic government. The current government is fully democratically elected. The People's Republic of China, however, still views Taiwan as part of their territory. Taiwan's status as a nation, therefore, is somewhat in flux, with no nation wanting to anger China by fully recognizing an independent Taiwan.