What is a galaxy's bulge?
It's the central part of a galaxy, where there is a much higher concentration of older stars, dust, and other luminous objects.
The reason it is denoted a bulge, is because the high luminosity stands out from the rest of the galaxy and looks like it is bulging outwards. Here's an artistic rendition that is probably a bit inaccurate, but illustrates the concept because it is fairly exaggerated.
It is also commonly accepted that most galaxies contain a supermassive black hole which resides in this central bulge.