Apples are Ryuk's favorite food and probably the only thing he eats. He likes apples from the human world because they are "so juicy." Ryuk stated that apples are an addiction for him, like alcohol or cigarettes for humans. He has only been seen eating red apples (likely fuji apples). Ryuk exhibits symptoms of withdrawal if he goes without eating apples for a while. These symptoms include contorting his body into uncomfortable positions and desperation to the point that he will take orders for one (i.e. looking for hidden cameras in Light's room).
Apples also symbolize the 'forbidden fruit' because it is generally thought to be the fruit mentioned in the biblical book of Genesis in which the snake told Eve that eating the fruit from the Tree of Knowledge of Good and Evil would give her the same status as God. Light's goal is to become a 'god', therefore confirming this theory. However, Ohba said in an interview for Death Note 13: How to Read that he chose apples because he thought it was "cool."
Apples have become a very popular commodity or currency in the Shinigami Realm when Ryuk returns with an unknown quantity after the end of the series. In the manga one-shot, Midora bribes the Shinigami King with 13 apples she brought back from the human world in order to receive an extra Death Note, which she gives to C-Kira.
Gukku, one of the first Shinigami seen in the series, comments in the one-shot that Ryuk bringing human apples to the Shinigami Realm is the reason why Death Notes become so easily available. Gukku also states that "he doesn't know if that's a good thing or not." This could possibly be a precursor to more Death Notes being brought to the human world by Shinigami in future possible storylines.
Apples from the Shinigami Realm look like shriveled green peppers and do not resemble normal apples at all. Misa gave a juicy human apple to Ryuk and in return, he lets Misa have a bite of an apple from the Shinigami realm which Ryuk was carrying with him at the time. Misa takes a bite of the apple and then spits it out in disgust saying that it "tastes like sand".