For a few centuries people have cut brains up and named the various lumps, bumps, holes, and sheets that they found. This could be compared to the early days of astronomy, just the naming of stars, without any reference to what a star is or why it shines. Let us call this desire to name bits of the brain cerebonomy, just to have the joy of coining a new word. Cerebonomy is simply naming the brain parts.
More recently, with the aid of microscopes and other techniques, researchers have been able to discern a multitude of layers, regions and divisions within each lump or bump that had previously been given a name. Each layer and region is then also given a name, or possibly a number. This makes the simple profusion of names one of the chief obstacles to reading brain-related literature, and there is no way I can think of to simplify things.
Here are some bits you should know in a simple glossary.
The brain stem is the bit of your brain that is obviously an extension of your spinal cord. It is not divided into 'left' and 'right' sections like the rest of the brain. On a side view of a human brain you can see just a little bit of the brain stem sticking out the bottom like the stem of a cauliflower, but it carries on up inside almost to the centre of the brain, and indeed incorporates some (or all) of what is usually called the mid-brain or mesencephalon.
The cerebral cortex (confusingly often just called 'cortex') is the folded 'cauliflower' sheet on the outside that is the most visible feature of a human brain. It isn't very thick (cortex is from the latin for 'bark' of trees), in humans it is around 4mm thick. It is divided into the left and right hemispheres although these are joined in various places by large, fast, bundles of connections. This folded surface sheet is the 'greyest' part of the brain indicating that its connections are dense and mostly very short-range.
The next bit of interest, the diencephelon, refers to most of the bits that you cannot see from the outside (because they are covered by the cortex) and which are joined to the top of the brain stem and the mid-brain. If you turn the brain to look directly up the brain stem, you can catch a glimpse of part of the diencephalon.
An obvious feature of the human brain is the cerebellum sticking out the back near the bottom. It is almost like a separate little brain and is joined to the brain stem. Its surface is folded - more narrowly folded than the cauliflower cerebral cortex - and consists of a single sheet of tissue in a folded arrangement like an accordion.
These are the major divisions of the brain - not really like the divisions of Gaul, more like the continents of the globe. Like the continents there aren't too many of them.