How do non-price determinants of demand affect a product's status as a normal or inferior good?
Nonprice determinants of demand are the only factor that determines whether something is an inferior good or a normal good. Specifically, it is customer tastes that determine this.
An inferior good is one whose demand increases as consumers become poorer. For most goods, an decrease in consumer income would bring a decrease in demand as consumers were able to afford less of the product. But with inferior goods, poorer customers buy them more and so their demand increases as incomes drop. These would be things like goods from Wal-Mart as compared to goods from a more "high class" store.
For the most part, a good is inferior if people think it is inferior. Consumer tastes determine which kinds of clothes and foods are more desirable. These become normal goods while the goods that are seen as less desirable become inferior goods.