Sometimes when a book (or other form of media) completely rocks my world, I think about what core elements inspired that reaction from me and what I can learn from it that I can apply in my own writing. On rare occasions I'll admire a creator's skills in setting and worldbuilding and artistic medium choices (word artistry in books, art style in visual media), but first and foremost, a work that grabs me is almost always about the characters.
So I just put together a short list of five things I enjoy seeing in characters--things that make me stay for their stories and hang onto their words.
1. They have a past.
Good characters, for me, never feel like they started living when the author began typing page one. They have a past, and the fact that they have a past is clear from how they act and interact (even if we, the consumers of the media, do not know what that past is). They have grown from birth to their current position in life affecting and being affected by their reality, and you can see evidence of that. And it must not be one-dimensional, like a disaster defining the entirety of who a person is as a trauma victim or a person who thinks of nothing but revenge. Their past must be complicated and formative, and must be woven into who the character is.
2. They have a future.
Everybody I want to hear stories about wants something. They're going somewhere. If they aren't sure where they're going, they have something to say about that, too. Good characters are being framed in the context of their media at what is theoretically the most interesting part of their life--when they're going from now to later, for better or for worse, and they are showing us how they're moving forward. Good characters imagine their futures and have opinions or aspirations about getting there.
3. They have relationships.
Stories are rarely about one person who doesn't interact with others (and has never done so). Good characters should have history and current feelings regarding other characters and those relationships should be complicated. That is not to say they have to be messy or negative; it's just that organic relationships in real life are an amalgam of many factors, and fictional relationships should be similar. Authors should know (though not necessarily reveal) how every important character met every other important character, what they like and dislike about each other, what important things they have done together, and what they want from each other. And the real skill comes in (hand in hand with item 1 above) when authors make us feel what those relationships are in the spaces between the words without spelling them out necessarily (though sometimes explicit explaining happens).
4. They have quirks, attitudes, opinions, thoughts, and actions.
In real life, people have inside jokes; they have gestures; they have favorite foods; they have hobbies; they have political affiliations (sometimes); they have religions (sometimes); they think and do certain things in their lives that reflect what's going on in their heads. A good character always has a mental life as well as a set of actions and words we can observe. And even if we are not partial to that mental life, evidence of it must exist. That is not to say people have to fall into traditional boxes or that all opinions have to be consistent with other opinions (cheese knows we all know inconsistent people in real life!), but people have to internally make sense and externally reflect that they are alive. They might have speech patterns that differentiate them from other characters, or fashion preferences, or always wear a certain item of jewelry, or have a talent or a challenge or an allergy. These parts of good characters flow naturally from who they are as people; they're never just collections of attributes stuck together on a stick figure. Good characters have these quirks, etc., and they make sense with what we learn about them in the story.
5. They change.
Hand in hand with number 2 above, as good characters go toward their futures, they are changed by what happens to them, change themselves, or try to change themselves. Even very settled, established characters who exude stability by not changing very much in a story still need to demonstrate that they can learn, or might be changed in small ways by helping others change more radically. Completely static characters aren't just boring; they aren't realistic as people. We love seeing people change, even if it's not for the better and can't be called inspirational. We tune in for other people's stories because we want to watch them move from one place in their personal journeys to the next, and we are unlikely to enjoy the ride if the travelers learn nothing and end up back where they started.
There are other less generalizable elements of characters that I tend to personally latch onto, but these are the boiled-down simplified versions of character traits I can identify in pieces that have moved me in extraordinary ways. If I'm just not into a story, there's probably no one in it who feels real to me, and the above five items are what I think makes a character feel like a person. If I believe in a character, I might care about them even if I don't like them, and once that happens, I'll likely tune in to watch them do whatever they're doing . . . no matter what it is.