A bullet journal is simply a notebook in which one maintains an integrated calendar, journal, and sundry lists. In order to make this easier to manage, most people use an index and shorthand symbols.
The current idea of a bullet journal was developed, and is marketed, by Ryder Carroll. He has a very specific system of what should go where, what shorthand symbols you should use, and what components should be included. However, this sort of thing is neither new nor difficult, and the primary guidelines that make a bullet journal noteworthy are simply that you should keep things short, organized, and easy.
The current popularity of bullet journals seems to be a result of most people's rather dull idea of what organization looks like, and a deep-rooted belief that indexes are boring. I hate these people, and am glad that they are starting to see the light. Currently Instagram is full of people showing off their artistically decorated, personally optimized bulleted lists, as a new generation of people realize that even organization can be fun.