The Brain Twist is an odd puzzle. The four tips of the puzzle in its tetrahedron shape can be twisted. You can fold back the three pieces of a face and the whole puzzle forms a new tetrahedron on which the corners can also be twisted. The image below shows the brain twist slightly unfolded.
The puzzle is pretty easy to solve being equivalent to a Rainbow Cube. If you've solved a Rainbow cube before you'll probably remember how you could solve it a piece at a time with no tricky last layer cases. If you solve the three pieces of a single face, then the 6 pieces adjacent to that face (they form a sort of middle layer if you put the solved face on top), the last 3 pieces of your 'bottom layer' can be solved just by twisting a single vertex into position. It is possible to end up with 2 pieces that need to be swapped (not possible on this puzzle) in which case you will have started the puzzle with faces in the wrong order and need to start again.