These caramelize apples are meant to be slightly sweet, not over-tart and thoroughly warming. They are wonderful addition to ice cream or cracker, or just eat them alone! :)
You just need 4 ingredient, which most likely you already had it : apple, sugar, cinnamon and butter.
Melt butter in a medium skillet over medium heat, throw (well, throw gently, of course) sliced apples in the pan, I use granny smith. Sauté for couple minutes, then add sugar, stirring frequently until they just get tender. Sprinkle cinnamon over the cooked apples, toss the mixture gently for another 2 minutes, until the sugar begin caramelized and the apples are crisp-tender.
There's no precise recipe for this, just cook according to your feeling. You can try it with brown sugar, which I am sure is ummhhh.. or you may splash lemon juice over it for more tangy flavor. Again, go by your feeling :)