My granddad who was an immigrant from an island off the West coast of Connacht, Ireland, would tell me all about the fairy folk and the myths and legends that had been passed down to him. Taught me some things in Irish, too. He was homeless in the late years of his life (because of mental illness, then later, dementia) so he basically lived in the Borders bookstore. None of his family received any education past the 3rd grade, but he was extremely literate and autodidactic, so he was always giving me specific books on the topic of Irish folklore and culture -- and the paranormal beliefs. The notable most story he had told me was about Tir na nÓg, (land of the young) which was a phantom island (further) off the Wild Atlantic Way, one which was shrouded in fog and only appeared visible at rare times, one which was one of many portals to the Otherworld. The fae could, if you were unfortunate or seduced enough, bring you the island, where you could experience all kinds of sublime and ineffable pleasures and dreams with the fair hosts. Yet even an hour spent on the island would be like a hundred years when you return, with everything and everyone you had ever known gone and dead. He told me all kinds of stuff about the fairies, and I learned more from those books. It's worth noting that the popular image of "fairies" is dead wrong. This is not Tinkerbell. These creatures are chaotic, unpredictable, mischeivous, and often dangerously malevolent. You wanted to avoid fucking with the fairies or encountering them as much as humanly possible. In fact, that is the entire heritage of the reasons for the custom: actions and ritual gestures to prevent encounters with the fairies because they were such a serious and pernicious threat to daily livelihood.
Putting aside the cartoon Disney version in modern understanding, the actual fairies as actually understood by the modern Celtic cultures (Ireland, Scotland, Wales, Breton), are almost identical in descriptions of the encounters (which is how the myth spread, after all) as to UFO's and alien abductions. Down to the last detail. The only thing that is different is of course, the cultural lens and symbols through which it is seen or interpreted in the zeitgeist. But the time dilation, the complete abduction of random farmers or wayward travelers only to drop them in some bizarre place and condition, perhaps days or years later (with aging differences, even), the bizarre and inhuman figures described, the dream-like nature of the encounter, things in the sky, strange little men, weather events heralding an encounter, tampering with biological material including crops and human fetuses, alternating malevolence or almost divine graciousness, and everything else. What's different is whether they're encountered in an indigenous 10th century Irish island or whether they are encountered in 20th century America in the midst of a technological revolution and a burgeoning mass media. Of course the event and indeed, the myth, would be interpreted through whatever happens to be the dominant cultural paradigm and way of comprehending such an incomprehensible thing. And even today many older people in Ireland, especially in the rural West which up until the 90's was much further behind the transformation to an industrialized country spreading out from Dublin and Leinster, take the fair folk at least moderately seriously. Remember: they are called "the fair folk" not because they are so "fair" (as in good, light) but because they are always listening. That was the level the culture was at. They are "fairies" because to call them otherwise would be a grave offense for creatures who lurk anywhere to snatch you up and away.
Anyway, my grandad knew as much as there was to know about this stuff, and he respected the culture of the homeland he left. He was naturally a skeptic, but a very literate one at that. I wouldn't say that he strictly "believed" in any of that, although Catholicism and the Church was important to him and his family, as well as my grandma.
My dad on the other hand -- an innately skeptical, strictly empiricist ( yet still open when it counts I think) PhD. biologist -- from a young age, I would get into long debates with him where he would argue that lucid dreaming was "scientifically imossible" and "cultish New Age hogwash" even though I was trying to show him the recent-ish groundbreaking sleep science by Stephen LaBerge and more important tell him my own experiences having been a natural and frequent lucid dreamer for literally all of my life and describing that to him. He would not believe it on the grounds that there would be no way of knowing whether you actually experienced it or simply imagined it and reconstructed it upon waking. I understood then and understand now the logical parameters of that argument, but it's bullshit. Stephen LaBerge in the 70's had trained dreamers hooked up to EEG machines and eyelid monitors giving coded eye signals in real time to produce a stimulus from inside their dream to demonstrate this. But I really felt like the kind of skepticism my father was giving me then was the worst kind -- precisely because it was so useless. It was a falsehood. He was telling me that he did not believe the veracity of my experience for, really, no other reason than he himself had not experienced it. And then telling himself that it was his logical imperative to do so. That seems to be the worst kind of "skepticism": what is really just materialism and reduction -- I cannot see therefore I do not believe. It's not complex thought, it is stupidly simple: "I cannot admit the possibility of your experience because I cannot see it myself."
Anyway, sorry for rambling, but I enjoyed hearing the other responses here.