Rapa Nui, commonly known in English as Easter Island, is a volcanic island in the south eastern Pacific ocean, and one of the most isolated inhabited islands on earth. It was first settled by Polynesians some time between 300 and 1200 CE, who in their geographic isolation developed a unique variant of Polynesian culture, which after a statue building phase, entered a final pre/early contact phase that featured a religious trial by ordeal, the birdman contest by which they reserved scarce wild bird eggs for the clan of that year's winning birdman. Aside from their iconic statues (an outsize variant of wider Polynesian culture), their shortage of wood and abundance of workable stone prompted a strongly lithic culture which, despite the depredations of a century of sheep farmers who rearranged much of the archaeology into stone walls, and late twentieth century airport builders who reused much of the worked stone from one end of the island as hardcore for the island's unusually long runway, has left us a large archaeological legacy including petroglyphs (especially of vulvas), unusual dryland agricultural adaptations, fortified cave complexes, and the aforementioned statues.
Apart from a possible visit by pirates, Rapa Nui's first contact with European "civilization" was a visit by the Dutch on the 5th of April 1722. Without the consent of the islanders the Dutch attempted to rename Rapa Nui after a religious festival they were celebrating, an act of disrespect that still has repercussions to this day. Due to a combination of western diseases, slave raids, civil war, famines and possibly ecological degradation, its inhabitants, the Rapa Nui, suffered a population decline of 99% or more in the first century and a half of European contact. They have since partially reversed the population decline, but their contemporary culture has been greatly altered by the demographic bottleneck and Christianisation that came close to extirpating their unusual culture.
Its isolation, distinctive large Moai statues, mysterious unique and untranslatable Rongorongo writings, lack of historical records, iconic status in popular culture, dramatically changing scientific theories, and cultural definition of incest that severely restricts marriage choices among the Rapa Nui, have made it an irresistible lure for pseudo historians, pseudo archaeologists, slave raiders, missionaries and sundry adventurers.
Some of the irrational pseudo archaeology associated with the island started out as genuine scientific hypothesis but these were continued, even expanded upon, even after newer finds had disproved the original hypothesis. Others rely on a selective use of material that has the effect of both denying the ingenuity of the Rapa Nui, and of downplaying or even denying the horrific crimes carried out against them, and then there's the extraterrestrial stuff.
 The sunken continent
To the nineteenth century eye Easter Island was a mysterious treeless place with a vast quarry containing hundreds of abandoned incomplete statues, wide stone roads leading down the island and into the deeps below and ignorant locals whose explanation of the movement of the statues was that they walked from the quarry to their appointed sites across the island.
This all fitted together as the theory of Mu, the great lost southern continent whose only remains were a small island which was once a mountain peak with a large statue quarry.
Mu and the idea of Rapa Nui as being the remnant of a larger submerged inhabited land is now deprecated as a scientific theory, because while the island would have been larger during the low sea levels of the last ice age, sea level is no longer believed to have altered during the time it has been inhabited:
In the 1930s Metraux visited the island and identified the "roads" as natural lava flows.
Mapping of the toppled statues had established that the current coastline is almost circled by statue complexes (ahus), indicating that the statues were likely to have been placed when sea levels were much as they are today.
The main statue-producing quarry is now considered to have been demarcated into many different clan areas, littered with hundreds of incomplete statues abandoned after encountering inclusions of harder rock. While the previous concept of the quarry as one vast production line implied far more statues than are known, the modern interpretation of the site is not inconsistent with the number of known statues, and a sunken continent is no longer required to explain the size of the quarry.
 Were the statue builders Polynesian?
Proximity to megalithic cultures in South America, combined with theories of a lost southern continent and now largely abandoned ideas of "primitive" and "advanced" peoples prompted various speculations as to whether the contemporary Polynesian inhabitants of the island were the descendants of the statue builders, or whether the original statue builders had been supplanted by a later Polynesian migration. Evidence for the South American case included similarities of stoneworking technique between the island and various Inca and pre-inca sites, and biological links such as the sweet potato and the reeds which grow in lakes on the island and in Lake Titicaca in the Andes.
The Routledge expedition of 1914/5 excavated a number of the partially buried statues, and discovered patterns of marking that are lost from the exposed statues. Mrs E Scoresby Routledge was able to make a number of water colour drawings of elderly islanders in the islands leper colony. These watercolours of the tattooed posteriors of elderly Rapa Nui, and her observation of the similarity of these designs to the markings on the backs of the excavated statues was and remains a key proof of cultural continuity between the statue builders and the modern Rapa Nui (as the missionaries had stopped tattooing in 1862 during the Christianisation of the island this appears to have been the last opportunity for a scientist to make such records). She wrote a popular book about her voyage, which gives some detail of her scientific findings, along with her facing down a native revolt and her encounter with the German Pacific fleet at the outbreak of WWI. But unfortunately due to reasons of insanity she was never able to publish a full scientific report of her excavations and other research.
In the 1930s Metraux was able to compare the Inca and Rapa Nui stonemasonry, and identified that a supposedly similar style was actually quite different with the Inca shaping whole blocks to precision fits and the Rapa Nui only having done a casing around rubble filled walls.
Not withstanding this, Thor Heyerdahl led a large expedition to the island in the 1950s to promote his theories of early oceanic travel. Though several of the archaeologists in his team did genuine scientific work, his promotion of the South American theory does not help his own scientific reputation. Though it did sell a lot of books.
Subsequent biological analysis has established the species of reed on the island as a different, though physically very similar species to the one growing in Lake Titicaca. Whilst DNA analysis has confirmed the link between the modern islanders the statue builders and the rest of Polynesia.
The sweet potato is still considered by biologists to be a South American species, and therefore an indication of early and undocumented communication between Polynesia and South America. However its ubiquity in Polynesian agriculture combined with Rapa Nui's late settlement and subsequent isolation would tend to indicate that the sweet potato had reached Polynesia before the settlement of Rapa Nui.
The Rapa Nui people may have been one of the few civilisations on earth to have independently invented writing, or they may not. That they had the Rongorongo script before the 1860s is not disputed. Whether it was true writing or a series of memory aids used by chanters is in dispute. It is also not known whether the script predated the European contact era or was invented in response to a Spanish visit in the 18th century when the islanders were "persuaded" to sign a treaty in Spanish. The script's symbols are stylistically similar but not the same as some of the petroglyphs which adorn the islands rocks and statues,
The script is known today from a small number of incised wooden objects. As far as dating is concerned, three of the surviving samples of the script have been dated to the post-contact era, either by carbon dating or because they were carved on European style oars.
Some of the Rongorongo objects which we have today may have been carved in the late nineteenth century when, as now, mementos of the island were one of its few exports. At the time, and unlike today, there was no regulation over the sale, export, or even provenance of "ancestral relics".
What is generally agreed is that the last people who used the script died either directly from the large Peruvian slave raids of 1861/1862 or in a subsequent smallpox epidemic which killed most of the survivors.
The island's lack of trees was a problem to early chroniclers as they couldn't see how the statues could have been moved round the island without the use of wood for levers and rollers, and the islanders' own explanation that the statues walked was thought unhelpful (the vast majority of the statues are carved from tuff, an easily worked rock that only occurs in one quarry on the island).
This has naturally attracted the interest of pseudo historians and others with theories of exotic machinery.
Pollen analysis of the sediments in the crater lakes has now established that the island was densely forested with a relative of the chilean wine palm, this pollen declines after 1200 CE and disappears in 1650. Further evidence for these trees includes impressions from past volcanic eruptions and rat-chewed nuts found in caves.
This and evidence of episodic famine on the island has prompted more recent eco-collapse theories along the lines of "they cut down all the trees to move statues". Though prevailing scientific theories are that the trees were unable to reproduce due to rats eating the germinating nuts as well possibly as islanders cutting trees down.   
But it has enabled archaeologists and others to run various tests as to the best way to transport concrete blocks of the size and weight of a Moai. After various experiments with rollers, travois, and levers, the prevailing theory is that the islanders simply used ropes and a rocking motion, so that just as one moves a fridge around a kitchen, the statues were walked across the island.
 ZOMG they eat the island!
We have no reliable records as to when Hotu Matua landed with his two canoes on the beach at Anakena (and that isn't the only founding legend, but he does still have a shop on the island). We do have carbon-14 dates that establish the islands were inhabited by 1200 CE, which ties in with the pollen analysis and would give 450 years for the island to have been deforested, but the island could have been inhabited for many centuries before the earliest confirmed date.
The island is very isolated from the rest of Polynesia and at the time of European contact didn't have the pigs and dogs that usually accompanied Polynesian settlers. So it has been theorised that there may only have been the one settlement party, which was sufficiently ill-prepared or short of supplies that they ate their dogs and pigs, and only had chickens and rats by the time they landed.
There are various theories as to when first settlement occurred based on linguistic divergence, and settlement patterns elsewhere in Polynesia. We know that it came after the sweet potato reached Polynesia, but we don't know when or how that happened.
Ecological collapse theories tend to prefer the 1200 date over earlier ones, which coincidentally gives less than 700 years for the island to go from pristine wilderness to almost uninhabited and treeless. Paleoclimatic, historical, and cultural information indicate that the island was a little dry for normal Polynesian agriculture and despite dryland adaptations such as lithic mulch, the islanders suffered periodic crop failures. This would have been exacerbated by the loss of deep-sea resources after they ran out of trees to make canoes. However, linking the deforestation with population decline is contentious because such theories tend to ignore or understate the known and somewhat documented major population drops due to the slave raids and epidemics that the islanders suffered, and instead give undue weight to famine.