The Damsel (In Distress) is one of the more controversial roles in fiction, for many reasons, not the least of which being how it treats its characters. But we really should dig into what the character is about before getting into that!
Index
What, Where, Why
History Lesson
Good, Bad, and Ugly
Example Time
What, Where, Why
To put it simply, the damsel in distress is a character who needs to be rescued. This needs to be for a big enough reason and a long enough period of time that their rescue becomes a significant part of the plot, often even the defining goal or drive of the hero’s journey.
Princess Peach, from the Super Mario franchise
Princess Leia, from the original Star Wars trilogy, among others-
Lois Lane, from the Superman comic franchise, among others 
Mamoru Chiba/Tuxedo Mask, from the Sailor Moon franchise
Aurora, from Disney’s Sleeping Beauty movie and associated media
In most cases, the damsel’s primary function is to set a story in motion or to give the heroes a goal in their story. Lois Lane gets captured by Lex Luthor, giving Superman motivation to fight him, if the whole “saving metropolis” bit wasn’t enough already.
History Lesson
One can trace the damsel in distress as far back as Greek mythology — Andromeda’s rescue serves as one of Perseus’ most well-remembered stories. But even other cultures had their stories involving such things. Imagery in the Book of Revelations compares the Church to a woman in need of protection from a dragon, who is often interpreted as representing the devil.
The phrase “damsel in distress” comes from the french demoiselle en détresse, itself referring to the codes of chivalry of knights in the middle ages.
Good, Bad, and Ugly
The Damsel in Distress is one of the few tropes I would say is misused more often than not, simply by way of what the trope is. To put it mildly, the Damsel in Distress often acts as less of a character and more as a goal or prize. She’s the princess in the tower, whose hand in marriage is the prize for her recovery. She has no say in the matter, because she’s on the other side of the very scary dragon.
But there are a few ways to do this archetype justice. The most prominent one is seen with examples like Princess Leia; she may be a goal for a period of time, and she may even be captured several times, but ultimately she doesn’t just sit around when it’s time to get rescued. In order to give the character more agency and character, she becomes more than a prize, and has her own goals and needs.
The other way to work this archetype well is to work with it, rather than against it as before. For example, Fiona in the theatrical version of Shrek is given an entire song about the emotional trauma of waiting day-in and day-out for her knight in shining armor to arrive. While a less popular method, this archetype can be used to explore the emotions of feeling trapped and hopeless, and how people deal with that.
Example Time
The latest knight in a long line approached the tower. I sighed, leaning my head against the window.
“Another one bites the dust,” I say, absently, as the dragon roared. I could see it in my peripheral, but the awe had lost its charm after about the hundredth dead knight. I waved to the knight, yelling perhaps more bored than necessary, “Please, sir knight, save me from this beast.” I had always said it, so it was more of a formality at this point.
“I, Sir Hero, shall defeat you, foul beast, and save that fair maiden!” The knight said, charging.
I leaned out the window now to watch. “Good luck with that,” I said, as the dragon prepared its fiery breath.
That night, I wondered, absently, if I’d get more than a name from the next one.
Heroes, Villains, and Damsels, Oh My! 
I have a MULTIPLE thoughts about this trope and ways in which it is/is not sexist but for your sake I’ll try to keep it short.
1) STATISTICAL LIKELIHOOD of a female character being the one to get captured. Movies and TV shows really like to have just one or two female characters thrown in with a great big group of dudes. But here’s the thing – if there’s only one woman in your action team, and yet she’s consistently the one who gets captured? That’s not just sexist, that defies realistic odds (for example, remember how mad everyone was that Black Widow got captured by Ultron in Avengers 2? There were 5-to-1 odds that a captured teammate would end up being one of the men, but naturally that did not happen). In media where there’s actually an even distribution of male to female characters (or, gasp, more female characters than male! What?!), it seems way less bogus for one of the women to end up in distress, because there are better odds of that actually happening. In Orphan Black, there are about four female characters to every one male character, so it makes sense to see more instances of female characters needing to be saved. But in X-Men there are a lot more X-gentlemen than X-ladies, so it seems prettty far-fetched that the women are always the ones getting captured by baddies.
2) EQUAL OPPORTUNITY DISTRESS makes EVERYTHING BETTER. Princess Leia spends some time in need of rescue in Star Wars, but then in The Empire Strikes Back, Leia and Han end up having to bail Luke out of danger. And then Han gets frozen in carbonite and he has to get saved too! Grand rescues for all! Upping the stakes of who could end up in danger makes for a much more interesting viewing experience, in my opinion. No one ever worries about Fred or Shaggy in episodes of Scooby-Doo, and when a TV show gets predictable it gets boring. There’s a reason As You Like It is considered the crown jewel of Shakespeare’s comedies – everyone ends up in danger at some point, everyone gets to be the hero at some point, and it keeps the audience guessing! It makes the characters a lot more well-rounded as well, because there are lots of opportunities to showcase both their strengths AND their flaws.
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