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At first I was thinking I'd put this in the "weird" category. "Fantasy" also seems possible, and so does "romantic movies," but ultimately I think the best fit is "quirky."
Well, of course I wanted to see this because I'm a fan of Christina Ricci, who plays the title character, Penelope Wilhern (she also narrates the story). As it turned out, there were also some other familiar actors, including Catherine O'Hara, James McAvoy, Peter Dinklage, Burn Gorman (from Torchwood), Reese Witherspoon, and Russell Brand. Anyway, the story starts with Penelope explaining that several generations ago, her family was cursed, when one of her ancestors was planning to marry a servant girl, but his family made him abandon her and marry someone else. The woman he'd been going to marry "fell off a cliff" after that- I'm guessing it was suicide, but that wasn't really clear to me. So, her mother, who happened to be a witch, cursed the family that had shunned her daughter. The first girl to be born into the family would have a nose which resembled the snout of a pig, and this curse wouldn't be lifted until one of her own kind (meaning an aristocratic sort, apparently) accepted her for who she is. Well, generations went by, and only boys were born into the family. Penelope was the first girl since the curse was placed.
There were lots of reporters around when Penelope was born, because her parents were used to living very public lives. Apparently the possibility of the curse didn't occur to them. When Penelope was born, however, her mother (O'Hara) was horrified, and hid the baby away from society. One reporter, named Lemon (Dinklage), um... lost an eye in the process of trying to get a picture of the baby. Anyway, Penelope grew up in isolation, not allowed to leave the house or have any friends. Until the time she turned 18, at which point her mother started trying to marry her off, to break the curse. However, all her prospective suitors ran off as soon as they saw her face. Most notable among these was Edward Vanderman. Until him, the Wilherns would always make people sign non-disclosure agreements concerning Penelope's appearance, but Edward got away before that. (After him, the family would start having people sign the agreement before seeing Penelope, rather than after.) But Edward was so horrified of Penelope that his mind apparently played tricks on him; he saw her as a hideous monster with long, sharp fangs, who he was convinced wanted to kill and eat him. Seriously, that guy was ludicrously excitable and melodramatic. He immediately went to the police, but they didn't believe him, and a newspaper report came out that he had suffered a nervous breakdown.
When Edward went to the paper to demand a retraction, he met Lemon, who was the first person to believe him. They both wanted to restore their reputations, as Lemon's had been wrecked many years earlier, after Penelope was born, though he never got a really good look at her. Lemon wanted Edward to go back to take a picture of Penelope, but he was too scared. So they decided to find a down-on-his-luck blue blood, named Max Campion, who had, over the last few years, gambled away his family fortune. In spite of being poor, he still had the pedigree to break the curse, so they gave him $5000 to pose as a suitor, and get a picture of Penelope with a hidden camera. (When Lemon first approached him in the pub where he was playing poker, I had a suspicion which would much later turn out to be true, but I won't spoil it. It was a rather amusing twist, though.)
Anyway... at this point I want to say that it's just ridiculous how totally repulsed everyone, especially Edward, was by Penelope's appearance. Maybe I would have found it more startling to see her in person than in a movie, but still... it's not like her whole face was piglike, and even the snout wasn't a full-on pig snout. There was some degree of nose-ishness about it. Maybe I should also say that because I'm a sci-fi fan, I'm used to tempering my perception of aliens or supernatural creatures and such, so I'm able to see them as sort of "normal." But honestly, aside from the snout-nose, which admittedly wasn't attractive, the rest of her face was really quite pretty (it is Christina Ricci, after all). This, to me, is the single most unbelievable aspect of the movie... the degree of repulsion everyone felt.
Anyway, Max meets Penelope, and gets to know her a bit before seeing her (she would talk to him through a one-way mirror). They develop genuine affection for each other, but when he sees her, he regrets the deal he'd made with Lemon and Edward. This leads to a misunderstanding, and eventually Penelope decides to just run away from home, covering up the bottom half of her face with a scarf, and seeing the outside world for the first time, on her own. She makes a friend named Annie (Witherspoon). And um... well, there are other plot developments that I don't want to reveal (though I will say Lemon comes to abandon his obsession with the Penelope story). I should also mention that the curse is eventually lifted, in an unexpected way. (Which, btw, leads the mother to a realization about herself, which doesn't make sense to me, because I thought Penelope's father... Well, I can't really explain my question without spoiling a plot point I want to avoid. Sigh.)
But anyway, it all has a happy ending, and in spite of feeling early on that the movie was just too ridiculous, I eventually came to like it fairly well. It really was kind of cute and amusing and romantic and, you know, quirky... in addition to being totally ridiculous.