Hokkien-Taiwanese
Hokkien-Taiwanese refers to the Minnan language of southern Fujian. Due to the inconsistent usage of Chinese language terms to refer to both entire linguistic clades as well as the main language of the clade, Minnan in the broader sense is also used to refer to the entire Southern Min Chinese phylogenetic branch, which besides Minnan proper includes Chaoshanese (Teochew), Leizhounese, and Hainanese.
Neither Hokkien nor Taiwanese are the only language spoken in Fujian or Taiwan respectively, so Minnan can also be used to specify the language in certain contexts, to contrast against for example Taiwanese Hakka, Taiwanese Mandarin, the other Fujianese languages of the Min Chinese branch such as Eastern Fujianese (Fuzhounese), or the Formosan languages of Taiwan. In Southeast Asia, Hokkien refers to Minnan (Southern Fujianese) by default, rather than Eastern Fujianese.
There are many dialects of Hokkien around southern Fujian and Southeast Asia, and generally the prestige dialect of the 19th and 20th centuries was considered to be that of Xiamen, which formed as a blend of accents from Quanzhou and Zhangzhou prefectures. Before the rise of Xiamen, Quanzhou was the main city of southern Fujian, so traditionally the Quanzhou dialect was quite prestigious, and it is still the preferred dialect for Nanguan music. Taiwanese dialects and accents are formed from a blend of Quanzhou and Zhangzhou accents much like the Xiamen dialect is, but overall Taiwanese leans a bit more towards the Zhangzhou side of the dialect spectrum, especially the prestige accent of Tainan.
Hokkien-Taiwanese inter-intelligibility varies as one travels across the southern Fujian region, but overall, Xiamen is the most similar to general Taiwanese, which also has many Zhangzhou features. Within Taiwanese, the accent of Yilan is even more Zhangzhou-leaning, while the accent of the Lukang area is Quanzhou-leaning. In southern Fujian, as one moves inland westwards from Zhangzhou prefecture into Longyan prefecture, Hokkien starts to blend with the local Hakka lects, so by the time you reach Longyan's Xinluo district, where much of the population also has Hakka heritage, the local variety of Minnan has decreased intelligibility with mainstream Hokkien. Intelligibility also lowers as one reaches the northern bounds of the southern Fujian region in Datian county. Towards the northeast, Hokkien reaches its border with the Puxianese (莆仙語) language of Putian and Xianyou, also historically known as Hinghua (興化). Puxianese is phylogenetically part of the greater Southern Min branch, but diverged as it became phonologically influenced by Eastern Fujianese (Mindong). Southwardly, Hokkien of Zhao'an is somewhat transitional towards Chaoshanese (Teochew).
Chaoshanese is overall quite similar to Hokkien, and has some partial mutual intelligibility with it. Once the main rime isogloss differences between Chaoshanese and Hokkien are learned (these isoglosses also crisscross differently depending on which dialect of Chaoshanese and which dialect of Hokkien are being compared), intelligibility increases quite rapidly. The difference between Hokkien and Chaoshanese can be compared to the difference between Spanish and Portuguese.
Besides Chaoshanese proper, there is also Shanweinese (the language of the Hailufeng region) which exhibits many features of Zhangzhou-style Hokkien mixed with features of Chaoshanese, with certain rime isoglosses matching better with either Zhangzhou or Shantou depending on the particular accent. While the Shanwei prefecture is generally lumped with Shantou, Chaozhou, and Jieyang prefectures when talking about the greater Chaoshan region, the heavy Zhangzhou-descended Hokkien component as well as higher contact with Hakka and Cantonese has resulted in many considering themselves a separate identity from Chaoshan proper, though there seems to be an increasing sociolinguistic self-identifying with Chaoshanese and being part of the Chaoshan region in the younger generations.
Hokkien of the Southeast Asia diaspora can vary between Zhangzhou-leaning and Quanzhou-leaning on the isogloss spectrum, with some admixture from the similar Chaoshanese. For example, in general, Penang Hokkien rimes are Zhangzhou-leaning, and Singapore Hokkien rimes Xiamen-leaning. This is also complicated by the high level of vocabulary influence from Malay (both standard Malay and local varieties of Malay) or other local languages depending on the country, and code-switching in general between a number of Sinitic varieties as well as other local languages, the exact ratio of which depending on which languages are present in each particular locale.