REGINA Spektor is sinking into a sofa in a London hotel room, shoes discarded under the table in front of her, black-stockinged legs stretched out across its top. She's wearing a dress which, if it isn't already the national costume of some obscure European principality, deserves to be. And she's telling me about her school days in New Jersey as an immigrant Russian scholarship kid in a Jewish "yeshiva", or religious college.

"I felt I was just going to always live in a world of drones who only cared about their new pair of sneakers," she says. "I got the reputation of being that cray-zee Regina, that cray-zee girl'. I was, like, I'm not f***ing weird, but don't you think it's cool that there's a balloon floating away over there? Shouldn't we all look at that and take a moment to think about it?'" Her classmates didn't think so. Still, it's a winning image: the whimsical teen with the curly mop and a thing for Tchaikovsky pointing out beauty-in-the-everyday to dull, Nike-obsessed tenth-graders.

Now 29, Spektor continues to make observations like that. It's why the piano-playing singer-songwriter is often given labels such as the "Kooky Queen", and is compared to Tori Amos, another ivory-bashing eccentric.

The label is wrong, the comparison less so. There are more than 15 years between the two women but the common ground is easy enough to map. Like Amos, Spektor is a conservatory-trained pianist. Like Amos, she turns out intelligent, confessional lyrics in an extraordinary singing voice. Like Amos, she has what you might call a singular vision. But she's far more than just a noughties update of a 1990s archetype. Not that Spektor doesn't acknowledge the debt. Amos, along with Joni Mitchell and New York folkie Ani DiFranco, taught her that "girls" can "do stuff", as she puts it. What she means is this: make music, make records, win respect, tour the world.

Over the course of four albums, Spektor has done all of those things. She's in London this week to promote Far, her fifth full-lenth release. The high point has been an appearance on Later With Jools Holland, where, backed by a string quartet and seated at a grand piano, she performed Blue Lips, one of four tracks on Far produced by former ELO frontman Jeff Lynne. Among the three other producers who contributed is David Kahne, who also produced the last album by Spektor's mentors, The Strokes. More of them later.

Spektor returns to London at the end of the month to perform in Hyde Park, and in July she's at T in the Park, which she last played in 2006, the year she released her fourth album, Begin to Hope. Those are just the British dates. There are others. The itinerary is dizzying but, while she admits it isn't always easy, she still sees touring as a privilege.

"You don't have to scratch very far back in history to find a time when someone was being tossed around in a shaky horse-drawn carriage just to get to another town," she says. "Bach did it at an old age to hear one of his heroes play. People did it just to a see a cathedral, whereas we get to go anywhere. You can be in Scotland one day, New Zealand the next."

She has another reason to see this freedom of movement as a privilege. In 1989, when glasnost made Soviet emigration possible, some 70,000 Jews applied for exodus. Most went to the US. Spektor, then nine, was one of them, travelling with her parents - both classical musicians - via Italy and Austria. Home became a Jewish neighbourhood in the Bronx. Spektor lives in Manhattan now but her parents, Ilya and Bella, still live in the same apartment they've occupied since 1990. Spektor's room is untouched; still there if she wants it.

I ask her what she remembers about her Moscow childhood. "A lot," she says, reeling off a list of things: her music school, her regular school, her friends, her parents' friends, her parents' apartment, her parents' friends' apartments. "And I remember going with my mum to museums. They made you take off your shoes and gave you slippers because a lot of the museums were in old mansions. I remember sliding through these giant rooms, because you could go really fast, then wiping out next to some giant work of art. I remember going to the opera and ballet, too. All sorts of fun things. My parents took care to expose me to good culture."

I tell her I'm surprised her memories are so fond, especially given the difficulties her parents must have faced. She thinks for a few seconds before replying. "They're fond because for the most part, if the people who surround you when you're young are kind, then that's your childhood. You're protected, you're sheltered. I definitely have mixed feelings about Russia, though. There were a lot of amazing things that happened, but a lot of terrible things, too. Terrible amounts of anti-semitism, poverty, oppression. People just barely getting by and spending so much of their lives literally procuring, standing in line, trying to get things."

She remembers her mother coming home laden with shopping bags having spent hours queuing in some distant corner of Moscow. She remembers the need to plan everything out, but also the economic realities that made it impossible to plan. Food, was "just whatever was happening that day. Today it's meat? Cool. Tomorrow it might be frozen pineapple."

Spektor lives in an apartment which has no television. But she spends a good deal of time in hotel rooms that do - and, increasingly, they offer Russian-language channels. She is appalled for the most part at what the modern Russia has become. "They've taken all the negative things from pop culture and they're trying to do it in Russia. So even the really great things that were happening in the Soviet Union, like people not being materialistic and being interested in culture, are disappearing.

The pendulum has swung so far the other way that now it's all about how you look, how much money something costs. It's super-weird to see the two cultures collide."

What Russia needs most of all, she thinks, is a lucky break. "They've just had such shit luck with leaders that I think they're due a good one, even for a little while," she says. "Someone who actually cares and is not just there for the power. When was the last time they had someone who, even for just a brief time, actually gave a f*** about the people? I'm waiting. How many hundreds of years is it going to take?"

Spektor's grandfather arrived in America a year after the rest of the family. He was in his 70s at the time but he found the Hebrew and Yiddish he had been forced to abandon as a child returning to him. Spektor's parents knew only Russian but threw themselves into learning about a cultural and religious inheritance which had been banned under the Soviet system. In Moscow the family had secretly observed Passover, but knew only fragments of the lore and ritual.

"We knew to have chicken soup with matzo balls in it but we didn't know that you weren't supposed to eat bread. We knew little bits and pieces. We knew certain holidays but not others. When we came to America, my parents realised they only had a sort of telephone-game version of Judaism."

One thing they did know was that whatever anti-semitism they faced in the US would be a pale imitation of the discriminatory version practised by the Soviet state. "When I got to New York I was very aware that now we were free to be Jews," she says. "I didn't have to worry that I wouldn't get into university because I was a Jew. I had to worry that I wouldn't get into university because I sucked at maths."

What she didn't suck at was piano. Even before the family had one, even before they knew English, a teacher was found who would give classes for free. Her name was Sonia Vargas and she remains a close friend. Spektor still thanks her in the liner notes to every album.

The plan was for a career in classical music. It was the only life Spektor would countenance, the only thing she had ever wanted to do. In the Soviet Union her father had hoarded bootleg cassettes of The Beatles, Wings, The Moody Blues and Queen, but beyond that her musical world consisted of Tchaikovsky, Shostakovich and Prokofiev. When she left school she won another scholarship, to Purchase College in Westchester, New York. It was an arts college with conservatories covering music, dance, film and fine art. She was free to study piano full-time - and, even better, the students she met there understood the cool floating balloon thing.

"It was the place where I felt my mind was really freed and opened for the first time," she says. "I realised that there are a billion people like me. Everyone was expressing themselves, trying different things, being open to art, being excited about life and not getting bogged down with trends or any of that stuff you see around you in high school." She even managed a six-month stint in London, studying Shakespeare and French existentialism at Middlesex University in Tottenham. She couldn't not take a music class, though, so opted for 20th-century classical music. Ironically, it was in London that she first encountered the music of the New York-based composer John Cage.

Stockhausen, Messiaen and Boulez were also on the curriculum.

Gradually, though, she turned away from classical music. There was no eureka moment, just a realisation that she didn't have what she calls the "Olympian work ethic" needed to make it in that world. She began writing songs instead, frequenting open-mic spots in New York, meeting other musicians, opening up her work to criticism and - luckily, she says - being encouraged by "supportive people who didn't tell me how crappy my songs were". Everyone told her she could sing, too, which came as news to her.

She became a fixture on the so-called anti-folk scene, performing at New York venues such as the Sidewalk Cafe and the Knitting Factory, and selling her self-produced CDs. The first, 11:11, was released in 2001; the second, Songs, in 2002. Its cover shows Spektor wearing a Russian fur hat, a playful image she would reprise on later works.

One night, during a live spot, she played her song Poor Little Rich Boy, which involves Spektor accompanying herself on piano by bashing a drumstick on her stool. It deeply impressed a watching audience member, Gordon Raphael, who had recently produced The Strokes' debut album, Is This It. At the time, they were easily the hottest band on the planet. Raphael took Spektor into the studio and recorded Poor Little Rich Boy, the first song on what would become her 2004 album, Soviet Kitsch. She then toured with The Strokes on the release of their second album, Room on Fire, and later recorded with them.

Soviet Kitsch announced Spektor's arrival as a major new talent and built on her growing cult appeal. But it was its follow-up, Begin To Hope, which cracked the Billboard top 20 and put her on the international map. When the video for Fidelity, the first track from the album, debuted on YouTube it was viewed 200,000 times in two days. Not quite up to Susan Boyle standards, but not bad either.

Spektor's online presence includes YouTube tributes from star-struck fans covering songs including On The Radio (check out the hilarious ukulele version). The song's video proper, if you track it down, stars the primary-school class Spektor's mother teaches. (Formerly a music professor, Bella Spektor re-trained as a teacher after arriving in the US). Spektor herself is a regular visitor to her various fan sites, though it's necessity rather than narcissism that leads her there. She never writes down her lyrics or transcribes her music: it's all kept in her head and sometimes she forgets bits. Sometimes very large bits.

"It's lucky there's the internet," she says with a laugh. "People will transcribe a song I've only played once years ago, or even record it from a show, and put it on a website and I can go there myself and remember how to play it. Sometimes I've looked for set lists that way and found recordings and lyrics. Some of the lyrics are misheard, which is always really fun."

There's nothing ephemeral about Spektor's recorded output, however. She already has five albums under her belt and tells me she could have recorded five more if she hadn't spent the past three years touring. I can't see her ever wanting to climb off the piano stool - or see her growing community of fans letting her. And there's no more talk of cray-zee Regina. Now, when she points, a great many people turn to look.

Far will be released on June 23. Regina Spektor plays T in the Park on July 12