Quotation Marks ("") and Angle Quotation Marks (《》)
In mainland:

Both are correct. 

They are just two styles： ‘’ “” ,「」『』.

The former are borrowed from western countries, the usages in Chinese are identical with English.

The latter,「」 and 『』 are from Japanese. Although they are not so often used as the former, but they are definitely acceptable. In fact, Some people claims we should only use 「」『』 for quotation, because they are more suitable for Chinese character. They mean visual effects.

﹏﹏ is used in ancient texts acting likes 《》, they are for books, movies,songs and other specified names.

To 〖〗 ,don't know it.

The names of frequently used marks：

“” 引号，‘’ is 单引号，“” is 双引号。

「」 is called 直引号。

《》 书名号，专名号。

——  破折号。

；分号。

…… 省略号。

， 逗号。

。 句号。

？ 问号。

！ 感叹号 or 惊叹号。

You may find that in some technical writings, the 。 is  replaced by .