A Giffen good is an inferior good with the unique characteristic that an increase in price actually increases the quantity of the good that is demanded. This provides the unusual result of an upward sloping demand curve.
This happens because of the interactions of the income and substitution effects. Depending on whether the good is inferior or normal, the income effect can be positive or negative as the price of a good increases. Imagine an inferior good being Top Ramen (an inexpensive noodle dish, common among students). As your income rises, you actually consume less Top Ramen, because you may begin to buy more spaghetti, or steak, or something you enjoy more than Top Ramen. But if you lose your job, and your income goes down, you will consume more Top Ramen, because it is inexpensive.
Next we have to consider the substitution effect. No matter type of good, the substitution effect will be negative as the price of that good goes up. So if the price of Top Ramen rises, the substitution effect will dictate that you will buy more spaghetti, or steak because that good has become relatively cheaper.
The interesting thing about a giffen good, is that when the price of a giffen good rises, the income effect is greater than the substitution effect. So if a good is inferior, the income effect will be positive and larger than the negative value from the substitution effect.
Summary: If a good is inferior, a drop in income (represented by a price increase) increases the quantity of the good that is demanded. The substitution effect is negative for any good that experiences a price increase. A giffen good faces an upward sloping demand curve because the income effect dominates the substitution effect, meaning that quantity demanded increases as price rises.