Your bike actually has two bottom brackets - the first is the one that is actually the bottom bracket of your bike at the intersection of your down tube and your seat tube. It is the short, thick, metal tube that your cranks run through that is connected to the other tubes on your frame. Unlike every other tube on your bike, the bottom bracket does not get a different name than the part it holds inside it.
The part it holds inside it, is the bottom bracket... This consists of 2 cups which are pressed, or screw into the frame, along with a bearing that goes into each cup. With inexpensive bikes you have loose ball bearings that go into the cups and then there is some hardware that comes with the bottom bracket that keeps the bearings from falling out. In a 3-piece setup, the bearings and cups are all that make up the bottom bracket. Typically they may not even be called a bottom bracket, but just 'replacement cups and bearings' for your 3-piece cranks.
Smaller bottom brackets look cleaner and you're less likely to unintenionally hit them on a grind; but, since they have smaller bearings, they're more likely to break in the event that you do hit the BB.
One of the main reasons for anything being smaller is that you save weight. In the case of bottom brackets, you can shave over half a pound off the weight of the bike by using a Euro BB instead of an American BB. Euro is the small one and is threaded to make installation and removal of the BB easier. The American BB is larger and requires the cups to be pressed or pounded in.
Euro: Lighter, smaller, & threaded make it easier to work with. But, smaller means smaller bearings which can wear out faster (potentially), if the threads ever get stripped on the BB then you either hope the bike shop can fix it, or get a NEW frame! Be very careful with those threads people. Must use 2 or 3 piece cranks.
American: Larger size allows for use of 1 piece cranks and there are a ton of different bottom bracket sets available to use with it. The larger size allows larger bearings to be used which are more durable than smaller Euro BB. The larger size also increases weight and since it is not threaded, the cups must be pressed or pounded into the frame. This is not as precise as threading cups in, but there is no chance of cross threading.
Euro. There is very little proof that Euro BBs wear out quickly during bmx use. Even serious riders don't have major problems due to the smaller size. The weight that is saved on the bike is significant and the ease of installation/removal is all major factors in the decision. I also have enough confidence in my mechanical ability that I wouldn't strip out the bottom bracket installing it.
I have a spanish bb in my fly bikes pundai diablo 05 frame. They fit into a threaded euro bb in a frame. They basicly screw into the frame and stick out a tiny bit ( 2-3mm more than usual ). They just call it the spanish bb cuz it was intro duced my fly bike (wich are spanish). The bearings are bigger than a euro bb because they sit outside rather than inside the frame. see www.flybikes.com for pictures.