Easy Mode decreases the player's opponents' artificial intelligence and the level of their Pokémon. As such, battles become easier. This mode is unlocked after defeating the Champion in Pokémon White 2.
Challenge Mode increases some opponents' artificial intelligence and the level of their Pokémon. Some opponents will also have an increased number of Pokémon, as well as held items. This mode is unlocked after defeating the Champion in Pokémon Black 2.
In order to get Easy Mode, you have to beat White 2 on its Normal mode (which is its default mode), and in order to get Challenge Mode, you have to beat Black 2 on Normal mode. To get Challenge Mode on White 2 or Easy Mode on Black 2, you must use the Unova Link and swap keys with the respective editions.
When you activate Easy/ Challenge Mode, the Elite 4 and the Champion's difficulty will be changed accordingly the next time you fight them. However, to fight Gym leaders in these modes, you must restart your game (forfeiting your progress thus far), since the keys are only unlocked after you've gained the 8 badges.