The information about characters that ends up on the page takes many forms, does many jobs: it's in the narration, in the dialogue, it's in the exposition and action and tone. So the list below is only a few of the aspects of character that need to get onto the page.
Personally, a character description that gets down to the height of his/her cheekbones isn't important to me. It has its place, though. Some genres put far more weight on description, both of the characters and their world, because it's a cornerstone of the ambiance. Romances, for example. On the darker side of that, horror.
Are appearance and character connected? That taps into the old (and ever-relevant) question of whether beautiful = good and ugly = evil. And certainly if there's something unusual about a character's appearance, that will change how they're treated by those around them and that will influence their personality.
In general, a writer can say less than they believe they need to when it comes to backstory. But those details that are immediately relevant to the story -- that explain why a character has unusual motivations, or why s/he makes a choice that would seem odd to the reader -- do need to get onto the page. Not much more than that, though.
But it adds depth and complexity: up to a point, yes. This is another point where genre and reader expectations come into play. And your gut instincts.
The character's voice can be distinct from your narrative voice. If your character is the narrator, they're the same. Do you know who's narrating your story? (It doesn't have to be the character, even in tight third person.)
Voice is pervasive: it influences word choice, grammar, what's focused on and what isn't, what's glossed over and what's described in detail. A character's voice is influenced by everything from world-building to personality type. Well developed voices can distinguish characters from each other, within a story, using only a sentence or two of untagged dialogue.
In my experience, as a character develops you get a clearer sense of what's important to them, what they focus on, what words they use and why... all of which your gut uses to structure their voice, because your gut knows how people structure the things they say to emphasize what's important to them.
If your gut's having trouble with that, I bet that reading about how to write speeches or advertising copy would help. Those are both situations where information is structured specifically to get a certain idea across.
Action is character, after all.
Inactive characters are a personal peeve. It's a valid point that inaction defines character as well as action does -- I just find such characters annoying -- but let's talk about small actions for a minute. Let's say your character needs to chew on something, mentally, for a while and this is going to explain things the reader needs to know.
Your character could do this while sitting in a chair staring at a blank wall. Boring. You could say that the character is doing some simple chore while they're thinking. You could, also, intersperse the thinking with small details of the actions involved in that chore -- making them not generic actions but small indicators of personality.
Say your character's making a salad while they're ruminating. How do you make a salad? Personally, I take all the components out, pile them up and work my way through. I start with the lettuce, which I chop with a knife rather than hand-tearing. I don't use bagged mixes. These are all bits of my personality. (Feel free to steal them.)
What would you add to my list?