The App Store is part of the iTunes store (iTunes, app store, iBookstore). If you are getting a message to review your account when trying to download the app then have you logged into it and done so ? If you haven't already done so then you will probably need to enter payment details (even if the app is free or you have a balance on your account) - or you could see if this post by mountaingoatgirl lets you review your account without needing to enter credit card details : Re: What do I do if I do not have a credit card.
If you don't have a credit card, or don't want to enter it (you should be able to remove it after entering it, and the post by mountaingoatgirl doesn't work then to create a new account without giving payment details : Create an iTunes Store, App Store, or iBooks Store account without a credit card or other payment method - Apple Suppor…