If English is a language and American English a dialect of that language, then the equivalent comparison in the case of Taiwanese is that Southern Min is the language and Taiwanese a dialect of that language. Just as American English and British English are (mostly) intelligible to speakers of the other dialect, the same is true of Taiwanese and other varieties of Southern Min, such as Ē-mn̂g (Amoy/Xiamen) dialect and the versions of Hokkien spoken in Chinese communities around South East Asia.
Just as within American English there are further subdivisions of dialect and accent (Pittsburgh, Texan, New York etc.), so in Taiwanese there are different versions in different parts of the country. People from Gî-lân, Tâi-pak, Tâi-tiong, Lo̍k-káng and Tâi-lâm all have distinctive accents and in some cases different vocabulary than those in other areas of the country. Tâi-lâm is viewed as being the centre of Taiwanese language culture, indeed the city is home to both the National Museum of Taiwanese Literature and the 'prestige dialect' of Taiwanese.
Read more about speaking Taiwanese: