この means "this."
まち means "town" or "city."
について means "about."
どう is a "manner adverb" meaning "what way" or "which way" (according to a grammar book that I have).
思います is a polite way of saying 思う meaning "to think."
か is a particle that makes a sentence into a question.
このまちについてどう思いますか? means "What do you think about this town [city]?" どう (what way) in どう思いますか works as an adverb and is modifying the verb 思います (think).
All of my Japanese dictionaries say that どう is an adverb. I would say it can have different meanings according to how it's used and can be translated differently in English. For example, どう could be translated as "what," as in "What do you think? (どう思いますか?)," "how," as in "How was it? (どうでしたか?)," or "why? (どうしてですか?)."
This isn't exhaustive, but I hope this helps even a bit.