Neuroscience is the study of the nervous system. The nervous system is like an animal’s electrical system, transmitting signals throughout the body and neuroscience focuses on all aspects of how this system functions including the structure and development. Major branches of neuroscience include behavioral neuroscience (psychology), cognitive neuroscience (mental processes) and clinical neuroscience (neurology).
To learn more about neuroscience you shouldhave basic knowledge in the following areas:
1) Basic cell anatomy and function: neurons (nerve cells) are cells too! They do a lot of special, awesome things, but also some of the same things other cells do. This is very relevant in modern techniques used for studying neurons.
2) Basic chemistry, proteins: It's amazing but true: the transmission of information between neurons happens chemically! Whereas the signal (action potential) travels through a neuron, the actual exchange of information happens in the synaptic cleft. This is a chemical process that relies on various proteins.
Now that you've covered those subjects (they will probably be introduced in the first chapters of any good neuroscience book), you will need some basic knowledge of the brain and neurons. After this, you can dive into any specific that you find interesting.
Some of the common but most interesting topics in the neuroscience include
Brain and it's functions: What are the broad structures? What are their characteristics? How do brains differ between animals? What is different about the human brain?
Neurons: What does a neuron consist of? Are there different kinds of neurons? What is white/grey matter?
Interneurons
Neural organisation and Neural networks
Neurotransmitters: What is it, How many are there? How do they function in our body?
Neuroscience is HUGE, it consists of everything related to the brain and the nervous system, but the above should already give you a nice introduction. From there on, you can lean towards whatever you find more interesting. Some of them may be as follows:
Cognitive neuroscience: anything in (mostly) humans. Here the techniques are fMRI, EEG, TMS. They often try to explain more psychological or psychobiological phenomena, where the focus not so much lies on how it works on a neural level, but rather, where in the brain things happen. Although, it's a huge field that is more popular in mainstream media. I studied this and moved to more networks neuroscience, because I felt that there was a lack of evidence for some claims. None the less, it has done many many awesome things.
Computational neuroscience: Here, people use computational models to do various things related to neuroscience. Some people make networks that they use to make predictions, others do deep analysis of neural data. Then there are people working purely with math to make predictions. It's a hard field, but very exciting.
(experimental) Systems neuroscience: Overlapping with computational neuroscience. In this field people to recordings of neurons and analyse their activity to gain knowledge. That is as specific as I can make it, because associated techniques are: electrode recordings in animals (in order of popularity: mice, rats, drosophilia, fish, monkeys), optogenetics, genetics, microscopy. Much of the other fields also fall under systems, but this is roughly it.
Clinical neuroscience: A combination of any other field, but related to diseases