The main language is Mandarin Chinese. This sounds similar to the Chinese spoken in China, although with a different accent. However, the written language is different than China. China uses "simplified" characters, and Taiwan uses "traditional" characters. Traditional Chinese characters are the ones that were used for a long time before China switched to simplified, and they usually have more strokes (lines) or radicals (parts of the character). Simplified is usually fewer strokes and radicals. So, signs will be in traditional Chinese and you will almost always hear people speaking Mandarin.
There is also a local Taiwanese language called Hokkien. It is similar to Chinese in some ways, like a lot of the characters are the same and it has tones. But it is also very different. Aside from being pronounced completely differently, there are also a lot of characters that have no equivalent in Mandarin. It uses a lot of sounds that are not used in Mandarin, and has an entirely different grammar. Taiwanese is sometimes taught in school, but the younger generation definitely is less skilled in it than older generations. It is common that a family will decide that their child should focus on Mandarin and English and, therefore, lose skill in Taiwanese. However, it is very tied to the national identity. A lot of people have encouraged me to learn Taiwanese while I am in Taiwan, and have taught me a lot of words in it.
