Info-dump: The least preferred method. Info-dumping is simply telling the reader about the background information. This is usually pretty boring and stops the story dead in its tracks. You can use info-dump in little chunks if you're careful. Here's an example from my own writing:
She had saved them, thank the Twinned God above. As long as the Little Twins still lived, all was not lost.
All those generations ago when the First Twins, Phillip and Rose were born on the same day at the same hour to the leaders on opposite sides of a decades old civil war, the divided nation had seen it as a gift from the Twinned God, a chance to end the bloodshed at last. And so Phillip and Rose had married, united the people of Dnelend, and healed the wounds of years of conflict and hatred. They set forth the law to preserve the peace. The people loved them, and so it was decided that the Twins alone would rule over Dnelend forever. Each generation the new Twins were born to be taught and trained by their foreparents to reign in peace.
Sherrin nuzzled their soft fragrant baby hair. They were alive. That was all that mattered.
This is an example of a little chunk of info-dump at a point in the story where the action has slowed down, and it seemed like a natural time for the main character to be thinking about such things.
NEVER begin your story with info-dump. This is a fatal error.
Dialog: Dialog can be good or bad depending on how you use it. It should be dialog that your characters would actually say, not just characters talking about stuff they already know just for the benefit of the reader. (This happens all the time in Star Trek.) Here are a couple of examples:
"Don't do this," Hannah said. "It's dangerous." McKenzie looked at Jeremy and back at Hannah like a trapped animal. Jeremy only smirked.
"The invaders aren't just some fairy tale," Derek said. "You'll put us all in danger if you do this."
Jeremy's face reddened. "Why don't you go run for some leaders, encyclopedia boy? I'll bet mister smarty already knows which is the right key."
* * *
“You could ask Gavin,” Beatrix said.
“Gavin?” Her heart, already swollen with emotion, felt close to bursting. “I don’t think so,” she said softly. “I haven’t even seen him since . . . since I left.”
“Then he’d be perfect wouldn’t he? You can trust him, and no one has ever seen him in the Twins’ City.”
“I can’t ask Gavin. I just can’t.”
“Then who can you ask?” Mrs. Cuthbrite said. “Beatrix is right. He’s the perfect choice, and he would want to help you.”
Sherrin closed her eyes and pressed her cheek to Branwen’s. She could think of no other solution. She would have to ask Gavin.
Little bits of information: This is probably the best way to handle back story. Just drop little bits of info into the narrative, and trust your readers to figure things out without being told everything. Like this:
Pools of light from the nightlights lit the darkened hallway. One of them shone on a picture of the Salt Lake temple. She paused in front of it. It used to be world famous, she'd been taught. People came from everywhere to see it. She wished she could see it standing in defiance of the destruction of the invaders. Had she been told that or did she just imagine it? No one knew for sure what was Outside. Hannah had been in the refuge since she was only a few hours old. Even the adults didn't know the final outcome of the invasion.
* * *
Jeremy stood. Though Derek couldn't see his face, he could see the anger in his bearing. He clenched his fists at his sides. For a moment, a tingle of nostalgic fear overtook him. It reminded him of the time he'd stopped Jeremy from beating up Paul. Jeremy had stood just that same way then. Derek stood with his feet apart, his hands balled on his hips. I should have hit him then, and I should just hit him now.
These are just little pieces of background and characterization that give a whole lot of information.
Flashbacks: In my opinion, flashbacks are just as bad as info-dump. I don't have any examples, because I just don't use flashbacks. If you want to use a flashback, make sure it is interesting and shown in scene, rather than told in summary.
Prologues: I don't have an example of this either, but they can work pretty well, if you do it right. Again, make sure your prologue is interesting and shown, not told. One good example I can think of is the opening scene of Harry Potter and the Sorcerer's Stone. Rowling gives us a great, interesting little scene with a ton of information about the world.
Bottom line, getting the back story in can be tricky. It takes practice. I recommend taking a look at books you like and noticing how those authors have handled the back story.