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Padmé seems to fall in love with Anakin awfully quickly and in the films we never really seem to get given a concrete reason behind it.
Could it be that it's supposed to mimic the OT (in that Padmé sees Anakin's loyalty to her and his love for his friends throughout the film) or are we simply supposed to assume that they're having tons of important conversations off-screen?
In this "love featurette", the actress portraying Padmé, Natalie Portman speaks to the reasons why her character was attracted to Anakin;
The official novelisation also answers this quite neatly
He's young, handsome and she finds him physically attractive
He's dangerous (powerful and headstrong)
She senses his deep connection to the Force
She respects that he's devoted his life to helping others.
Her relationship with him represents a form of rebellion that she's never encountered before (forbidden love)
She looked over at Anakin, who was sleeping somewhat restlessly. She could see him now, not as a Jedi Padawan and her protector, but just as a young man. A handsome young man, and one whose actions repeatedly professed his love for her. A dangerous young man, to be sure, a Jedi who was thinking about things he should not. A man who was inevitably following the call of his heart above that of pragmatism and propriety. And all for her. Padmé couldn’t deny the attractiveness of that. She and Anakin were on similar roads of public service, she as a Senator, he as a Jedi Padawan, but he was showing rebellion against the present course...
She's also impressed with his openness and lack of guile about his feelings:
“You are in my very soul, tormenting me,” Anakin went on, not a bit of falseness in his tone. This was no ploy to garner any physical favors; this was honest and straightforward, refreshingly so to the woman who had spent most of her life being attended by handmaidens whose job it was to please and entertaining dignitaries whose agendas were never quite what they seemed.
And, of course you can never underestimate the power of a man who can make a woman laugh :-)
Padmé had to clutch at her belly, she was laughing so hard. Caught up in the whirlwind of the moment, Anakin sprang to his feet and ran off to the side, cutting in front of a shaak and frightening it with his sheer jubilance.
In the movies it's clear that he followed his heart, and she was very similar.
(heavily paraphrased)
Anakin: "I want to go to my mother and help her, but I can't, I'm a jedi and I've been ordered to stay here with you on Naboo."
Padme: "I'm going to Tatooine. You're supposed to protect me, but I'm going whether you go or not."
Yes, he was also good looking, powerful, and hung out with powerful people. He was also flexible, and when boundaries chafed, he found other outlets to resolve conflicts he had. They both had very similar situations where they are boxed in and bounded by various external forces, and both were creative in finding ways to resolve internal and external conflict while still maintaining their standards, or at least appearing to maintain their standards. They were both flexible idealists - both wanting peace and happiness for the universe, both working toward that end with all their ability in their different spheres.
And, finally, he absolutely, openly, and unabashedly adored her. It was he that first suggested they both leave their lives and be together while she resisted, then later as he became more bound up in his path she tried to convince him to do the same. Until he was caught in Palpatine's trap, he would have left everything behind for her. In fact this was the core of the trap - by binding her fate to a belief that Palpatine and the dark side could save her, Palpatine knew Anakin was his.
This, on top of the constant, close contact between the two, outside the reach and influence of others, gave him time to court her and help her develop feelings for him.
Valorum's answer seems pretty complete, but I just wanted to add that the Revenge of the Sith novelization offers some additional insight in chapter 9:
Her life before Anakin belonged to someone else, some lesser being to be pitied, some poor impoverished spirit who could never suspect how profoundly life should be lived.
Her real life began the first time she looked into Anakin Skywalker's eyes and found in there not the uncritical worship of little Annie from Tatooine, but the direct, unashamed, smoldering passion of a powerful Jedi: a young man, to be sure, but every centimeter a man—a man whose legend was already growing within the Jedi Order and beyond. A man who knew exactly what he wanted and was honest enough to simply ask for it; a man strong enough to unroll his deepest feelings before her without fear and without shame. A man who had loved her for a decade, with faithful and patient heart, while he waited for the act of destiny he was sure would someday open her own heart to the fire in his.
But though she loves her husband without reservation, love does not blind her to his faults. She is older than he, and wise enough to understand him better than he does himself. He is not a perfect man: he is prideful, and moody, and quick to anger—but these faults only make her love him the more, for his every flaw is more than balanced by the greatness within him, his capacity for joy and cleansing laughter, his extraordinary generosity of spirit, his passionate devotion not only to her but also in the service of every living being.
He is a wild creature who has come gently to her hand, a vine tiger purring against her cheek. Every softness of his touch, every kind glance or loving word is a small miracle in itself. How can she not be grateful for such gifts?
This is why she will not allow their marriage to become public knowledge. Her husband needs to be a Jedi. Saving people is what he was born for; to take that away from him would cripple every good thing in his troubled heart.
