What is the right use of bake and broil?
I look inside the oven and realize that when baking the heat comes from below and when broiling the heat comes from the top.
Although there's a intuition which says that the oven is small and so everything should be at the same temperature from the inside I'm sure that as I have two options there's at least one big difference.
What is this difference between them? Why do I bake a cake and not broil it? Which should I use for all kinds of meats? Depends? If I could extend this question to the default temperature of each, why are they different?
Broiling involves high heat coming from a source very close to the surface of the food. It is appropriate for thin food, or for adding color (via caramelization/Maillard reaction) to already (or mostly) cooked items. I couldn't say what the temperature of the heating element itself is, but the resulting air temperature is usually upwards of 550˚ F (288˚ C).
In baking, although the heating element is usually on one side or the other of the oven,* the idea is that heat is being transferred to the food more or less evenly from all sides. This is part of the reason thorough pre-heating is important -- to make the air in the box all the same temperature before the food goes in. There's no real "default temperature" -- you use whatever will produce the desired results in your food.
With lower heat, coming from all sides, baking will allow the interior of a thick piece of food to cook before the exterior has over-cooked. Putting cake batter under a broiler will likely result in a dried-out or burnt top, perhaps a layer of reasonably well-cooked cake, and a raw bottom. The heat simply won't transfer through the cake fast enough to allow the entire thing to cook.
The appropriate technique for meat depends on the size of the meat. Steaks can do very well under a broiler -- the intense heat allows the exterior to brown and become flavorful without the interior over-cooking. A roast cut or a whole bird, on the other hand, would need to be baked (or "roasted") so that the interior has time to come to the proper temperature. A very thick chop or steak might be started in the oven, then finished under the broiler to give it color.
*Your oven probably has just the one heating element, used for both baking and broiling. There's no reason that an oven couldn't have multiple elements for baking -- and it might produce more even cooking -- but broiling always means heat from above.