I listened to the Simple Minds' song Don't You. It has following chorus:
Don't you, forget about me
Don't, don't, don't, don't
Don't you, forget about me
Why is this usage of imperative mood correct? I thought that right usage is:
Don't forget about me
This is a turn of phrase that is used often in American English. Not only in the example given in comments. It's not even only "don't you" do something. Explicitly stating the subject you can add an imploring nature to a sentence.
Don't you even think about doing that.
Stop, you!
Hey, you!
All of the above mean the same thing, and imply "you" even if the explicit "you" is removed.
The pronoun you is not technically needed here, rather it is a colloquial addition used for directness and emphasis in the imperative mood. However, while in your example the sentence would make perfect sense without it, be careful, because there are many cases where the sentence itself is colloquial and wouldn't work without the you, such as 'don't you dare!'
As with most colloquialisms, there are no hard and fast rules as to how it's used, you just need to get used to it! Song lyrics are a very good place to start for this one, as they often implore someone else to do something!
Incidentally, this 'superfluous you' in the imperative has a long and complex usage in both American and British English. One finds it in a slightly different context in Shakespeare, for example in the speeches of the Welsh captain Fluellen from the play Henry V, whose many idiosyncrasies include the frequent interjection ''look you'', and whose overuse of the pronouns you, youself etc. were used to convey his verbose and bombastic language, which was a stereotype of the Welsh at the time (and to an extent even today). See for yourself here:
