When do you use two commas around extra information?

You'd use two commas around extra information when you need to! ;)

Putting "extra" information between commas is better known as using apposition or, more technically, a nonrestrictive appositive to modify what you were writing about, without changing the scope. Essentially, you're giving a different angle to something just written, perhaps identifying it a different way but with more info the second time around.

Example: Jake, an honorable chap if there ever was, traveled across town to return the wallet he found lying on the street. The wallet, which was stuffed with cash, would have seen a different fate if Gregory, Jake's dastardlybrother, had found it.

I'm an addict to parentheses, but using THOSE is akin to talking out the side of your mouth like some sort of wiseguy. When you want to tell things straight and fill in some extra info, the appositive can help you without calling much attention to what you're up to (not so with parentheses, ya see).