Sympathetic characters are vital in fiction. Developing sympathy helps us like them, root for them, and connect to them. If the reader can't sympathize with a character, they won't want to spend hours and hours with them and their story!

But what does that mean exactly? How do you make a character sympathetic?

For the most part, readers will like a character for the same reasons they like people in real life. For example:

They act unselfishly.They are kind to others.They are humble.They are (through no fault of their own) the underdog or a victim.They try hard to achieve their goals.They don't give up hope.They are brave.They are honest.They are clever.This is not an exhaustive list of course. It is also not a list of Everything A Character Must Have. But generally speaking, the more of the above traits a character has, the more readers will like them. (Careful though: it's possible to create characters who feel too perfect.)
The inverse is also true: readers tend to dislike characters who are selfish, mean, arrogant, self-victimizing, apathetic, hopeless, cowardly, deceitful, foolish, etc.
But creating sympathetic characters is an art. A character can be foolish but unselfish and kind, or they can feel hopeless but still try to achieve their goals anyway, etc., and we may still like them. Depending on how it's done, we may even like them more as a result of their flaws.
For example, Frodo gives up hope, but we root for him anyway (partially because Sam doesn't give up on Frodo either). Naruto is a fool, but we cheer when he doesn't give up. Han Solo is arrogant, but dang if we don't love him for being brave and (eventually) unselfish.

And what about anti-heroes and villains? Can they be sympathetic? Heck yeah! Being sympathetic doesn't necessarily mean we agree with or even like the character; it means we understand them and why they do what they do.

Like, Zuko is an arrogant jerk, but when we learn that his reason for chasing the Avatar is to return home and prove himself in his father's eyes, we sympathize. Magneto lies, kills, and has given up hope entirely on the human race, but given how mutants are treated in his world (and how he was treated in Nazi Germany), we can understand why. Killmonger is a lying, psychotic killer, but he also has some very solid points...

There's nothing wrong with a villain who has no redeemable qualities whatsoever, but a villain we sympathize with—with whom we find ourselves almost agreeing if it weren't for their methods—can be really powerful and even unsettling.

Again, this isn't an exhaustive list of traits that make a character likeable, but it's a strong start. It's also a good list to check against if you're getting feedback that people can't connect with your characters or just don't like them. Readers may not always know why they bounce off a character, but there's always a reason.
Who are some of your favorite characters (whether heroes, anti-heroes, or villains)? What about them makes them likeable to you?