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.