You ask a great question, but there isn't really one answer. Depending on how you look at it, there can be infinitely many different branches. Generally today, four main traditional branches of engineering are considered the core: chemical engineering, civil engineering, mechanical engineering, and electrical engineering. But the unique thing about engineering is that it never stays the same. Engineers are out at the forefront of new technology, and every time engineers begin combining their expertise with a new system or new technology new branches of engineering may develop.
You can check out our Try On a Career page for examples of some of the widely recognized branches of engineering, but even this is not an exhaustive list, and often engineers hold more than one title. For example, someone may be both a mechanical engineer and a naval engineer depending on the type of work she is doing, and an agriculture engineer may actually describe herself as a biological engineer. Often these "branches" are just descriptors to try to describe the work or the type of technologies an engineer may work with. Since there is no end to new technology there is also no end to the different types of engineers that may exist,
I hope that helps answer your question, and hopefully it give you even more questions to think about!