More specifically, I want to know what is the single quotation(') in 'bout means. I know 'bout means about but does that mean we always can use a single quotation for words that start with a vowel? Or is it just some specific words?
It's not a "single quotation", it's an apostrophe, which is often used to indicate omitted letters.
'Bout' is not standard English, but occurs in some dialects.
There are some cases where two different forms of a word exist in standard English, such as round/around, and waken/awaken. In these, both are normal words, so there is not an omitted letter, and the shorter version is not written with an apostrophe.
But when a shortened version is not standard, then if people want to write it to reflect the way it is being spoken, they will usually write an apostrophe. You will often see this in dialogue when the ending -ing is pronounced as -in (non-standard, but common). This is usually written -in', with an apostrophe.