In Taiwan they speak

1) Taiwanese mandarin (not exactly the same as in PRC, quite similar though).
… Chiang Kai-shek's army moved into Taiwan … those soldiers and refugees were from all provinces of China where they spoke different languages such as Cantonese, Shanghai, or Santong. Owing to the linguistic diversity and national language policy, most of mainlanders have switched from their first languages to Mandarin Chinese. Therefore, the Mandarin language is generally regarded as the lingua franca among the Mainlanders.

2) Taiwanese dialects: Hakka, Minnan (=Holo) and others
… generally speaking, there are four primary ethnic groups: aborigines (1.7%), Holo (73.3%), Hakka (12%), and Mainlanders (13%). … they have different ethnic languages, … Holooe, Hakfa, and Taiwan Mandarin, not mutually intelligible.

The language policy in Taiwan … reflects the mono-lingualism of the colonial governments. … only a particular language was chosen as the official or national language. Other languages were denounced as "fang-yen (=dialects)".
http://www.geocities.com/taigibun/taibun/ch2/ch2.htm

However:
Mainlanders are those who came to Taiwan after 1945. Those Fukkien and Hakka who settled in Taiwan before the Japanese occupation are usually called Taiwanese or native Taiwanese. Fukkien are the biggest group among them. Exactly speaking, these Fukkien were mostly from the south of Fujian province which is called Min-nan. So it often happens that the term "Taiwanese" simply means Fukkien descendants and the Taiwanese language means the Min-nan language. http://www.tufs.ac.jp/ts/personal/ogasawara/paper/epaper1.ht...

I prefer to use “Taiwanese language” to refer to Taiwanese Mandarin Chinese, and “Taiwanese dialects” for the ethnic languages.

Cheers,
Paolo