Why isn't "thing" OK here (despite being listed as one of the translations)?
This is just something you have to learn. Usually, before a vowel, it's "an" in English, but with a vowel that sounds like a consonant it's "a". "U" in "useful" is pronounced like the word "you". Not "oo", as in "moon". So it's a vowel sounding like a consonant - use "a".
I have "an" instead "a" before useful? I put it like an useful and duolingo says that is wrong...
Because useful is pronounced as if it starts with a consonant, hence it's a useful.