Zingiber officinale is "grocery store" ginger. By "ginger" variety standards, it is kind of a plain ginger with non-descript green bracts. This naturally goes dormant, so it would be easy enough to grow it in a pot and bring it in for the winter.

You might find galangal(Alpinia galanga) in a Thai store.

Hi Lisa - I too am planning to grow some grocery store bought ginger. Found some today for $3.99lb and got two enormous rhizome clumps - going to plant them tonight. See the posting on here titled "Planting Store Bought Ginger - there is a link there with lots of information - just all the information I was looking for!
Good luck!

I have grown it quite successfully in my garden after starting it in pots in the house in late winter/early spring. You need to plant the rhizome on it's site just like an iris. Keep it most and warm...moisture is key as if it dries out it may revert to dormancy or even die.
Good luck.