For those who didn't grow up in Victorian Europe and are otherwise unaware of this particular "science", phrenology is the determination of a person's personality and mental capacity according to the shape of the skull. Phrenology earned a fair bit of popularity in the early 1800s, but is now (rightly) shunned by the psych and neuroscience communities.
The idea behind phrenology stems from the notion that mental functions are localized to discrete areas of the brain. German physiologist Franz Joseph Gall's interest in the area was piqued at an early age when he noticed that among his classmates, those with prominent eyes also excelled at verbal memory tasks. He deduced that the frontal lobes were the centre for verbal memory and suggested that the size of a brain region is directly connected to its functional strength. From there it followed that stronger (and therefore larger) areas would create bulges in the skull to match the corresponding bulges in the underlying brain tissue. Using a map of brain function and scalp massage to determine cranial topology, an individual's personality could be assessed. And thus was born phrenology (then called craniology).
It was Gall's student, Johann Spurzheim, who really helped the spread of phrenology. He popularized it in the English speaking world and, notably, America, where one of the first Phrenological Societies was established. Spurzheim switched the focus from the anatomical to the social and political applications, appealing to those looking for guidance, and offering people a way to change through mental exercise.
Interest in the technique picked up, largely due to the innate human desire to understand - and predict - oneself. We can still see this today with the inordinate interest in things like fortune cookies, psychic readings, palmistry and astrology. The trend then, as now, was for man to try to rationally explain his own behaviours. Gall's craniology provided a tool understand, predict and maybe even control human behaviour and its popularity began to snowball.
In the US, popularity was booming. Journals such as the American Phrenological Journal granted legitimacy to the practice, earning praise from the likes of Thomas Edison and Alfred Russell Wallage, and the businesses popping up popularized it through talented salesmanship. The Fowler brothers turned phrenology into big business in the tradition of the great snake-oil men, with an entertaining and riveting show advising individuals on all aspects of their lives, from love lives to employment prospects, and ending each show with a practical demonstration. Phrenology had established itself both as a science and in the public mind as a legitimate industry.
Obviously, the shine of phrenology didn't last forever - it certainly isn't an acceptable specialty at any medical school I'm aware of. It was a combination of things that contributed to its downfall. While phrenology was enjoying popularity in North America, it was coming under attack across the pond. One of the primary figures challenging the science was Peter Mark Roget, the English physician famous for his contributions to Encycolopedia Britannica and the eponymous Roget's Thesaurus. Roget, and others, disputed phrenology on both methodological and physiological grounds.
The first objection raised was against the very underpinnings of the phrenological movement. While the field was predicated on skull topology being directly related to brain topology, there was no evidence of a connection between the two. In fact, whether 'high energy' (i.e. more developed) parts of the brain were quantitatively larger was the object of dispute, nor could the hypothetical sub-organs being proposed by phrenologists to explain their science be observed on dissection. On top of physiological arguments, phrenology fell prey to the classic fallacy of equating correlation with causation.
Phrenology was also under assualt from outside the scientific community, with religious leaders taking aim. Central to the phrenological 'science' was materialism, which is in contrast to the church's position of mind-body dualism. That is, the idea that personality and behaviour is strictly a consequence of the physical attributes of the brain is contrary to teachings of a soul. As a result, phrenology was condemned as implicit atheism. This, coupled with growing rejection in the scientific community and waning novelty of the practice led to the disappearance of phrenolgy and its being discarded as a credible science.
Now, there are other pseudosciences that have a similar history to phrenology, yet they endure. Astrology, for example, was widely accepted as reality. It was (and continues to be) opposed by religious leaders and has been rejected as a real science. However, astrology columns can still be found in any newspaper. Likewise, you can still get a tarot card reading, have your palm read or visit a psychic very easily. How is it that they persist but phrenology did not? There are a couple of reasons that might be.
First is its short history. Compared to the other pseudoscience mentioned above, phrenology was a flash in the pan. It may not have had time to establish itself in the public consciousness in the same way that astrology was able to. Furthermore, despite the poor methodology and basis in reality, phrenology was a scientific endeavor. Being such, results couldn't be rationalized with occult-based explanations. If predictions were wrong, the theory was invalidated. Phrenology was pushed out in favour of other ideas of self, such as the psychoanalysis of Freud or Jung. More importantly, some aspects of phrenology were absorbed into the body of scientific thought, making it easier to discard the invalid ideas. Modern neuroscience obviously still holds a material view of the mind, but the idea of localized functions and specialty areas - the basis of phrenology - have also survived. Every fMRI lighting up the music centre of the brain, for example, owes itself in part to phrenology - the brain science of its day. (Some have even suggested that functionl imaging is the phrenology of the 21st century) Despite having gone the way of the dodo, the study of phrenology remains an interesting piece of science history whose influence can still be seen.