There are two ways: cold and dryness. Rotting is caused by fungus, which are living things and require chemistry to work. This chemistry is slower the lower the temperature, so keeping apples cold will slow down the rate at which they rot. Unfortunately there is a limit to how slow you can make this without freezing the apple, and freezing the apple does other bad things to it (ice crystals break apart cell walls, which turns frozen fruit into mush).

Dryness also works but is harder to accomplish. In this case, the sugars get so concentrated that any fungus trying to invade and rot the apple tissue will have all of the water sucked out of its cells, which of course kills the fungus. However, this means that you've got dried apples, not fresh apples.

Rot is caused by a fungus. The fungus reproduces by spores from the air, so there are a few ways to reduce rot. First is to keep the apples cool. Fungi do not grow well in cold environments. The second is to keep the apple sealed in a container to prevent fungal spores from attaching. Also, broken apple skin will allow fungus to get in more easily. These two suggestions are covered by putting the apple in a refrigerator! It's probably the best way of preventing rot. If you have an apple that has already been cut, it may turn brown. This isn't actually rot, it is oxidation (reactions with oxygen in the air). This is the same reaction that causes guacamole to darken at the surface and iron to rust. It isn't generally harmful to eat, but may ruin the taste. To prevent them from turning brown once cut, the best thing to do is simply eat them quickly. Otherwise, keeping them in a tightly sealed container with little air can slow the oxidation.

Apples rot like any other plant material, exposure to oxygen and moisture enhance bacterial/fungal growth. If you are referring to the browning of apples, that is a different story. Apples brown when their tissue is exposed to oxygen. It is a chemical process called oxidation and it is harmless. You can easily eat a browned apple. One way to inhibit this is with lemon juice. Lemon juice is acidic and prevents the oxidation of the apple slices. It also give it a nice sweet and sour taste!