Store cards are cards that allow you to purchase items on credit at that store (and nowhere else). They are often tied to the store loyalty program. In many ways they are similar to credit cards but they are administered by a different type of organization and have different rules. These are not to be confused with store branded actual credit cards (cards that can be used at any store). It seems to me that this type of credit was more common in bygone days when real credit cards were not as common and people shopped a lot at one place. Store cards will often give great rewards if you shop a lot at that store.
When the article mentions "cash-back cards," it's referring to regular credit cards. Many high-quality credit cards in the US offer some kind of rewards---miles, points, or cash-back---as a percentage of what you spend on the card. Examples are the Capital One Quicksilver card and Chase Freedom Unlimited card, both which gives 1.5% cash back.