What is that hook (or J) looking thing below my bathroom sink?

One of the great parts about plumbing is that a lot of times the names don't actually make sense.  The hook/J under the sink is actually called a P trap and you'll find them in most fixtures that actually take water (and stuff) away from the house.  What it does is create a water barrier to keep air from coming back into the fixture.  At the bottom, there is water on both sides of the hook which blocks air (and aroma) from coming back into the house.

A good example of when it doesn't work is when you go to a summer cottage or maybe upstairs in your grandparents house.  In both cases, the toilet has not been flushed in a while (or the sink has not been used) and the water has evaporated leaving a distinct aroma in the air.  Once you run the water, it usually goes away because the water barrier is put back in place.

Have a great day!