Siamese cats probably originally came from Siam, or Thailand as it's known today, though I confess I never saw one there during the five years I lived in the kingdom. (To be more accurate, I saw hundreds and hundreds of Siamese/Thai cats when I was in Thailand, but they weren't the long elegant pampered darlings I was familiar with from Canada. They were spotty or striped skinny street cats with wily eyes and kinks in their tails.)
As a breed, one of the Siamese's most distinctive characteristics is its colouring - dark tail, ears, feet, and lower face; it's called "seal point", and has been present in cats from Siam since ancient times, at least according to a 16th century Siamese drawing of a feline with this colouring. Beyond this, though, the present-day Siamese breed does not much resemble ancient Siamese cats, which were rather round in face and body with squinty crossed eyes and kinked tails. (Cats with kinked tails are common throughout Thailand.) The modern Siamese breed has had its squint mostly, and its kink totally, bred out, and it has also acquired a long lean body with an elongated, almost triangular face by being crossbred with other breeds; they are of course also bred for their point colourings. Modern Siamese have blue eyes and a piercing, yowling meow.
Interestingly, Siamese are actually dark-coloured cats with a gene which inhibits the pigmentation of their fur in areas where their body temperature is higher. When kittens are born they are hot all over, and hence pale, but as they grow their extremities cool and develop the characteristic dark pigmentation; as the animal ages, it will become darker all over. If kitty gets a sore paw that is bandaged for a while, that limb will emerge paler than the others because of the heat generated by the bandage. Cats with a fever will also turn a bit paler.
Common tales of "Oriental" barbarity relate that the Siamese cat was worshipped in Siam, that theft of a cat was punishable by death, and that only the king owned any. Not true. However, they do seem to have been a common favourite of royalty and frequent residents of royal wat (Buddhist temples). In fact, most wat in Thailand today have resident cats and dogs, as devotees make merit by leaving food in the temple grounds for unfortunate homeless animals. If you want to get rid of a pet in Thailand, take it to the wat; it will have a home of sorts there, and a steady diet of jasmine rice and fish sauce.
Although it is sometimes said that farang - westerners - were not allowed to own a Siamese cat or remove one from Siam, they did find their way into European zoos and were displayed at the first British cat show in 1871. But in those days a Siamese was not a breed of cat, but rather just a cat from Siam, so many Siamese cats that were taken to the west in the early days probably didn't look much like the ones we associate with the label today. Siam, like other countries, had (and has) lots of different colours and shapes of cats.
The first documented export of a seal point Siamese was a gift given to the American First Lady by her consul in Siam; this poor female, Siam, was sent from Bangkok in 1878 and only reached of the White House the following year, where she soon became ill and died. In 1884, the British vice-consul Edward Blencowe Gould allegedly received a pair of Siamese from the king Chulalongkorn; the story is that the king offered him anything his heart desired, and was dismayed when Gould chose the cats. However, most historians dismiss this story as romantic drivel concocted to increase the glamour of the breed and drive up their price, and it seems more likely that Gould bought these cats at a market. In any case, he sent them to his sister, who bred them and showed the kittens in an 1885 cat show at the Crystal Palace, where they won "Best Short-Haired Cat" and "Best Cat in Show". Any embargo that might have existed was then swept away by the huge demand that was generated for the enchanting felines, which were known as "the Royal Cat of Siam".
Today Siamese are very popular with cat breeders and fanciers alike, and they come with many different point colours. Because modern Siamese have been bred to have an extreme body-type, there has recently been a move to return to the more rounded head and body shape of the original cat from Siam.
Cat World: A Feline Encyclopedia, by Desmond Morris.