Two goods that are independent have a zero cross price elasticity of demand : as the price of good Y rises, the demand for good X stays constant
Independent goods are goods that have a zero cross elasticity of demand. Changes in the price of one good will have no effect on the demand of an independent good. For example, a person's demand for nails is usually independent of his or her demand for bread, since they are two unrelated types of goods.
Note that this concept is subjective and depends on the consumer's personal value function. If a certain consumer uses nails to cut bread, then for this specific consumer, nail and bread are not independent but rather complementary goods. If a certain consumer likes to eat nails, then for this specific consumer, nail and bread are substitute goods. These are extreme examples but they come to stress the point that the type of good depends on the subjective value function of the consumer.