The river running under Ember gets top billing in the first chapter, as one of the big reasons Lina doesn't want to get a Pipeworks position on Assignment Day. As she puts it, "A swift underground river ran through the Pipeworks, and every now and then someone fell into it and was lost" (1.37). Yeah, that does not sound safe.
But the thing is, Lina has never seen the river before in her life. She just knows about it. Which reminds us of the fact that just about everyone in Ember is only partially aware of the things that are central to their lives. They know the sky is always dark, but they don't know it's because of the fact that they're living underground. They know that electricity works, they just don't know how. They know they can grow things in the greenhouses by applying light and water, but they don't know why life can come from a seed.
The fact that the people of Ember don't actually know how they're surviving puts them in a tight spot. It's not like they're living in some jungle paradise where delicious tropical fruits grow on every tree and birds practically fall from the sky ready to be roasted and eaten. So they rely on the river for power, and use the generator to get that power. But if and when the generator finally breaks down, they won't know how to rig another solution in order to continue living in their city. By making Ember dependent on the river, but not telling them how exactly that dependence works, the Builders gave them a raw deal.
Doon has also never seen the river—until he takes Lina's spot as a Pipeworks laborer, that is. On his first day on the job, he's absolutely floored:
He stood still, staring. Like most people, he had never really been sure what a river was—just that it was water that somehow flowed on its own. He'd imagined it would be like the clear, narrow stream that came out of the kitchen faucet, only bigger, and horizontal instead of vertical. But this was something entirely different—not a stream of water, but endless tons of it pouring by. Wide as the widest street in Ember, churning and dipping and swirling, the river roared past, its turbulent surface like black, liquid glass scattered with flecks of light. Doon had never seen anything that moved so fast, and he had never heard such a thunderous, heart-stopping roar. (3.10)
Life in Ember is so contained, with everything planned out by the Builders, that it can feel a little sterile at times. But the river doesn't fit inside a tidy box, just like it doesn't fit in Doon's expectations. In a way, the river symbolizes all of nature, which they've been deprived of for so long. It's messy, wild, chaotic. Humans can work with it for a time, as the Builders did by harnessing the river's energy to power the generator that powers Ember, but you'd have to be power-crazy to think you could actually control it.
It's also worth noting that the river is the escape route from Ember. Lina and Doon are the ones to discover this, of course, and they're the first citizens to take a ride on that river—sans lifejackets. In that sense, the river represents hope to them, although it's not without its dangers. Then again, hope rarely is totally safe.