Generally in older fiction, and in more 'traditional' stories today, women prefer men who are strong, independent, and willing to make the first move. However, in many forms of modern fiction (and in Real Life), some women prefer men who are seen as 'weak' or vulnerable. This trope is about the latter.
A woman who finds weakness or submissive behavior inherently attractiveand a turn-on. Some women find that having the dominant position in a relationship (pun possibly intended) is a turn-on, as the 'power' is theirs and can be quite addictive. Also the healing that she's eager to give can lead to them falling in love with each other, creating the Florence Nightingale Effect, her dream come true.
To fit this trope, the desired or fetishized male must be seen as 'weak', either in temperament or body.
Temperament: Usually this involves a naturally submissive or sensitive personality on the man's part, or, if he's naturally aggressive and strong, a sense of vulnerability around the female which makes him 'weak' for her. Usually, the male will be younger than the female as well.
Body: Bishōnen may fall under this category, with their slender builds and feminine faces (though many bishonen are still badass). Ill Boys or injured men also fall under this, as do adolescents, because their weaker bodies indicate a vulnerability on their parts. Not surprisingly this overlaps with Tiny Guy, Huge Girl on occasion.
Compare All Girls Want Bad Boys, which shares similarities to this trope except that personality failings and anti-social behaviour as / rather than weaknesses are what attracts the woman to the man. See also Beastess and Amazonian Beauty, for the type of woman who an Amazon Chaser in this scenario is likely to be attracted to.
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There's an imaginary one (or three, or seven) in Peacemaker Kurogane. Every time one of the older guys imagines Tetsu in a relationship, it's either with him as an extreme Uke, or with an older woman who is the dom in the relationship.
Brutally used in Welcome to the NHK. Misaki selects Sato for her "project" of curing him from being a hikikomori, and the time they spend together eventually leads to there being quite a bit of romantic tension between the pair. It's later revealed that Misaki chose Sato, because she wanted to find someone who was even more "worthless" than she was and prove she could fix them because of her abusive stepfather telling her that she was a useless human being. As a broken recluse, Sato fit the bill.
In the Ranma ½ manga, Akane says this to Ranma to comfort him some after she finds out he's terrified of cats and becomes a frightened, submissive individual around them. Bonus points because she's tending to him at the time.
Akane: Don't let it bother you. It's cute to have a little weak spot.
In Axis Powers Hetalia, Hungary seems very happy with Non-Action Guy Austria. When his "vital regions" are being seized by Prussia, and he comes back beaten up, she jumps at the chance to fight Prussia off.
Kuroyukihime to Haruyuki in Accel World. She states that she finds the juxtaposition of his (potential) strength in the Accelerated World with his emotional sensitivity to be endearing and part of what drew her to him.
In a world of overmuscled macho superheroes, meek, helpless normalman finds he is irresistible to women partially because he's so pathetic that he activates some kind of motherly instinct and partially because he's the only man on Levram who isn't an oiled up thickhead.
Really more of a subversion. The guys may be shameless about playing this card when they're trying to chat women up, but the footage of the rugby matches shows they're anything but weak, and a couple (especially Zupan) are downright scary.
In Zombieland, Witchita actually commented that Colombus is like a scared bunny, or something along the lines. But he doesn't really get the girl until he goes all KILLER BUNNY on some zombies for her.
Frank Sinatra seemed to begin his movie career playing characters like these. Lampshaded in Take Me Out to the Ball Game.
P. G. Wodehouse liked this trope. In one book it is a strong trait of a girl one character likes, that it is part of her ambitions to be a good influence on the man she loves (a sort of Love Martyr syndrome), and he wins her over when she mistakenly believes him to be an alcoholic, so she stays by his side to save him from himself. (Which seems to be working, considering that the guy was not actually a heavy drinker.)
The phenomenon is further explained in Uncle Dynamite by the titular Uncle Fred whose nephew Pongo apparently elicits this reaction from girls (while not a strong or very heroic guy himself). (When Pongo's polar opposite Bill Oakshot, a large sturdy guy falls in love with the same girl he suggests that he'd be unable to use the same strategy and has to be the opposite, the "dominant male" - which eventually works for Bill.)
Another example: Jane Hubbard (big game hunter) and Eustace Hignett (poet) in The Girl on the Boat (aka Three Men and a Maid):
Jane was deeply stirred. Even as he sat, looking so pale and piteous, at the piano, her big heart had gone out to him, and now, in his moment of anguish, he seemed to bring to the surface everything that was best and most compassionate in her nature.
Miles Vorkosigan once lampshaded this trope himself. Being under five feet tall and, even after getting his bones replaced, delicate enough for most grown woman to readily overpower, he noted that he was singularly unthreatening and "it made them bold."
Angel: The possibility of intimacy with Angel doesn't comes up in Cordelia's mind until she glimpses a world in which she never joined the agency. This resulted in Angel inheriting Doyle's visions instead, causing him to go mad.
While Buffy tends to go for men who are much stronger than average, Dawn reveals in one episode, regarding a recently-injured Riley: "Oh, she just said you look even cuter when you're all weak and kitten-y, and she'd better go solo, or you'd get hurt."
She also quite enjoys nursing Angel back to health and in the Season 8 Comics we discover one of Buffy's sexual fantasies involves Angel and Spike chained to her, with Buffy dressed in a Naughty Nurse Outfit.
On 3rd Rock from the Sun, Dick spent an entire episode struggling to get a date with various attempts at being cool and manly. At the end of the episode, he discovers that talking about what a loser he is works much better.
Britta Perry on Community is a self-admitted example.
In Everybody Loves Raymond, Ray discovers acting utterly pathetic is apparently the most attractive thing a man can do for a woman. He uses this to finally get some alone time with Debra, and a friend of his uses it at the very end to successfully woo a co-worker he's had a crush on.
Ann Perkins, the nurse from Parks and Recreation, definitely has a Florence Nightingale thing going on. Being in relationships with stable, confident men freak her out, most notably her unhappiness with perfect physical specimen Chris Traeger. (It turns out though that Chris is super susceptible to flu season, and Ann is overjoyed to be able to take care of him.)
In Little Busters!, this can be assumed of the girls, given that the protagonist is an Ill Boy with a young-looking, feminine face and a kindly personality. However, Mio is definitely this: when he's Dragged into Drag, Kurugaya noted that she in particular was likely thinking that Riki looked Moe (which she unconvincingly denied) and later she claimed outright that she prefers feminine men.
In Something Positive, Kim has a thing for guys who aren't completely conscious, even going so far as to chloroform a few. When she's left alone to look after Davan, who's zonked out on pain meds, well... you can guess what happens. It's never treated as okay, but it still comes up for humor, years later.
Deconstructed in Homestuck. Vriska, resident backstabbingantiheroine is attracted to Shrinking VioletTavros...and it freaks her out. Most of the time, Vriska actively denies/represses it because even thinking Tavros might have any power over her disturbs her ego greatly. Even worse, she has no ability to express these feelings in a positive way, so her relationship to Tavros is less 'caring partner' and more 'abusive spouse'. (And Tavros isn't even aware that Vriska feels affection for him, given that she's the Poisonous Friend who once threw him off a cliff.)
Bones, the female bodybuilder in Cool Cat Studios, ends up dating the small, slender, and shy Michael.
In Justice League: Crisis on Two Earths, Superwoman is in a relationship with Owl Man, that universe's version of Batman. When Batman confronts her she seems to want to add him to the mix. Effortlessly breaking one of Batman's ribs seems to be a huge turn-on for her.
Kim's attraction to Ron in Kim Possible since she is much stronger then him.
In Aladdin: The Series, Prince Uncouthma's fiance Brawnhilda falls for Aladdin because he is smaller and weaker by comparison. When an assassination attempt leaves Uncouthma injured, she immediately returns to him, pledging to look after her "fragile darling".
In King of the Hill, after Dale knocks out John Redcorn who's trying to sneak into their bedroom after Nancy breaks up their affair, Nancy calls him up the next day to let him know she wants their relationship again.
Nancy: It's just, seeing you like that, passed out on the floor all vulnerable... like a wounded bear.