Prologues have fallen out of favor with a lot of people, and there's a good reason for that. Too often they are used to create a (false) sense of interest or urgency, or to throw in information that is meant to be important later on in the story, but can't really be shown anywhere else.
Even worse is when you read a prologue, get caught up in it, and then get snatched away into what seems a completely different story.
However, there are some prologues that are done in a way that. . .well, I hate to say, "correctly" or "properly", because those smell a little too much like RULES, and we're all aware how one sniff of that makes my inner contrarian rear up and start to screech like a banshee.
So let's just say there are some prologues that accomplish what any good opening does: it draws you into the story. It's not just tacked on as an afterthought. Or as a pre-thought.
Maybe this will work better if I "show" rather than "tell".
In Linda Lael Miller's historical romance High Country Bride, there are three rowdy cowboy brothers, and their father Angus decides they need to settle down. So he tells them the first one who gets a bride and fathers a child will inherit his huge ranch, AND the winner will essentially get to boss the other two brothers around.
While this is a great setup for a series, it seems like Angus is being a bit high-handed trying to run his sons' lives.
Until you read the prologue.
It starts with Angus at his wife's grave, updating her with details about their wild sons in order to justify his decision. It creates sympathy for this man, who is aging, and desperately missing his wife. But it also cleverly gives information about each son's personality and conflicts, and again, increases sympathy for Angus because he blames himself for their wayward lifestyles, and he wants them to settle down and be happy.
After a couple pages, I'm thinking, "Dagnabbit, they ARE a bunch of wild-mannered ruffians!" At the same time I'm mentally patting Angus on the shoulder, reassuring him that he did the best he could, raising three young boys while grieving the loss of his beloved wife.
This prologue is also a great example of giving description that is integral to the POV character, while showing who that character is. Here's the first sentence:
"Angus McKettrick hated every thorn and cactus, every sprig of sagebrush, every juniper tree and jackrabbit and hunk of red rock for fifty miles in all directions, and if he could have scorched the land bare as a pig's hide at rendering time, he'd have done it, yes, sir."
Another memorable prologue is from Suzanne Brockmann's Gone Too Far, which is Book 6 in her Navy SEAL Troubleshooter series. Sam is the hero of this book, although he's been an important character in each of the previous books (and Sam is a nickname he goes by, but his real name is Roger—it's information you need to know in just a minute).
In the previous books he's fallen in love with Alyssa, one of the other recurring characters, but she thinks he has dickish tendencies, and she won't get permanently involved with him because of that, and for her own emotional reasons.
Of course, Sam's dickish tendencies blossom when he is spurned by this woman he loves. However, at his core he is also heroic, which means he does wonderful things, and we know the dickish things are a result of emotional wounds he carries around.
The prologue of Gone Too Far starts with these words:
His lunchpail was empty.
Roger Starrett quickly closed it and latched it back up, looking around the middle school cafeteria, praying no one had seen.
Now who wouldn't feel sympathy for him after reading that?
In the next paragraph you find out that his mother is ill, and you can see his tough-guy façade as he informs the lunch lady that yes, he's eaten his lunch because look, his lunchbox is empty. These things would naturally create sympathy for any young boy, but they're doubly effective because readers have gotten to know Sam as an adult in previous books, and now there's a little more insight into the reasons for his behavior.
The prologue continues with the POV of Noah, a young boy who is being harassed by bullies. Sam comes to Noah's defense, and when one of the bullies ends up getting hurt, Sam does the heroic thing and provides first aid, even comforting the bully who is now really afraid and crying. As the prologue continues, you learn about Sam being bullied by his own father, and how Noah's parent informally "adopts" Sam, providing him the male role model that will steer him the right direction into manhood.
You can read the entire prologue on Amazon, and I suggest you do. There's a lot to be absorbed about deep POV, and how to show a character via another character's thoughts and actions. It's actually a wonderful story on its own merits. Even better, it makes you eager to turn the page and read more of this character's life.
Which is exactly what you want a prologue to do.