Basically, there are 2-3 components to any exposition: the information itself, the character thinking/talking about the information, and – in the case of information conveyed through dialogue rather than through the character's internal thoughts – the character recieving the information. Each character's reaction to the information – what does the conveying character think of the information, what is happening in the conveying character's immediate life that prompted him/her to think/talk about the information, how does the recieving character react to learning the information, what are they doing while the character conveys the information – is at least as important as the information itself.
John Scalzi's Old Man's War does this better than anyone else I've seen:
Chapter 2 exposits both space colonization and a recent nuclear war by having a prospective army recruit complain that his son died in the war against India, and yet India gets special treatment in sending settlers to space while Americans like him can only go as soldiers to die keeping the Indians alive. The main character points out that nuclear bombing killed millions of people in India and that large areas of the country are uninhabitable, but then the angry guy turns that around and says "Exactly! We won the war! Shouldn't that count for something?" The main character quickly decides he doesn't like this guy.
The same chapter then exposits the "science" of space elevators by having a retired engineer describe to the MC all of the reasons why the space elevator shouldn't work, then explaining that - since it works anyway - the space colonizing organization must be using advanced alien technology that they haven't shared with Earth scientists. He finally guesses that the space organization's grand gestures like the space elevator serve to protect them from interference by Earth governments: "if you can't figure out how we did this, then you're not capable of picking a fight with us."
A later chapter has a commanding officer showing the new recruits a slideshow of some of the species that they've met beyond the Solar System. The commander starts with a crustacean species that the MC reacts to as something out of a horror movie, then goes on to an anthropomorphic elk-like species that the MC associates with the wisdom of a mythical nature-spirit. The commander then explains that the crustacean species has produced some of the greatest artisans and mathematicians that humanity has ever seen, while a tribe of the elk species murdered an entire human settlement in the most sadistic ways imaginable. "If you don't get you stupid anthropocentric biases out of your head now, then you are going to get people killed."