Formation of the brain is quite a complex process.
It begins with a thickening of ectoderm (the outer layer of cells) above the notochord (which is a rod-shaped aggregate of cells that marks the future location of the spine). This thickened ectoderm forms the neural plate.
The sides of the neural plate fold up and eventually fuse - becoming the neural tube. The cells lining this tube are known as the neuroepithelium and produce all the future neurons.
The rostral (towards the head/nose) end of this neural tube forms the brain, and the rest of the tube forms the spinal cord.
The end of this tube which will become the brain begins to fold and expand. This forms three regions - prosencephalon (forebrain), mesencephalon (midbrain) and rhombencephalon (hindbrain). The most rostral end divides laterally to form the two lateral hemispheres of the cerebellum.
brain
There isn't a "popular name" for them. The most familiar chordates are the vertebrates, but not all chordates are vertebrates (tunicates, for example, are chordates).
lancelets
The Cranium
Chordates are that group of animals having some form of a spinal column (which include the vertebrates).
adult tunicate
All chordates are deuterostomes, meaning that in all chordates, the anus develops first during embryological development. All chordates are bilaterally symmetric. Most chordates have a complete digestive tract, and a body cavity. Note: not all chordates have these features, but it is believed that their absence in some chordates is secondary. All chordates are of course animals and eukaryotes.
Vertebrates are a subgroup of the chordates, meaning that all vertebrates are chordates, but not all chordates are vertebrates. Vertebrates are distinguished from other chordates by having backbones or spinal columns.All vertebrates have muscular systems that mostly consist of paired masses, as well as a central nervous system which is partly located inside the backbone (if one is present). The defining characteristic of a vertebrate is considered the backbone or spinal cord, a brain case, and an internal skeleton, but the latter is not true for lampreys, and the former is arguably present in some other chordates. Rather, all vertebrates are most easily distinguished from all other chordates by having a clearly identifiable head. Sensory organs--especially eyes--are concentrated at the foreend of the body, and there is pronounced cephalization.
Yes, most chordates are vertebrates.
Well, no they don't... There's a group of chordates called Agnatha which basically contains all the jawless chordates. And there are obviously limbless chordates present, the fishes and the snakes being the most well-known of the examples...
because
bilateral symmetry