Great drama has many dimensions (that's what makes it great), but the following are three simple things I see in all effective drama. [I think of these as pertaining to playwriting, but they could as easily pertain to screenwriting, or any form of storytelling.]
1) To make things big, make them small -
Details matter. Individual characters matter- they're the heart of the story. That is, if you want to write about the meaning of the universe, you don't write about the whole universe. You write about one or two (or three) people trying to make sense of it.
2) Every story is a ghost story -
Interesting characters are people haunted by something. Something, or someone, missing. Hamlet is the most obvious example, but everyone is missing something in their life- either that thing they lost, or the thing they can't attain (Willie Loman being the other obvious example). Often, it's the thing they can't fully see, the person from behind in a hat, the vague feeling of wanting something but not knowing exactly what it is. And you have to chase it. You have to look. It's the dark hole, the thing we peer into with infinite curiosity- infinite because it has no bounds.
3) Switch. And switch again -
This doesn't necessarily refer to the twist in a story (but it could, and the staple of mysteries). It refers more to moments, whether in speech or action, where a character or characters are taking us one place, and then suddenly shift gears- taking us in a new direction, darker or better or at least more interesting. A bad analogy would be a play on words (like The New Corker)- it makes you see it anew. The switch is something Sam Shepard talks about as a goal when working on dialogue. You don't always do it consciously, it just appears. And it's good.