The central conflict in Jeanne DuPrau's The City of Ember can be identified as a character vs. environment conflict.
In the story, the city of Ember was designed by scientists and builders as an underground refuge in hopes of preserving humanity in the case of an apocalyptic event such as a nuclear war. However, the builders designed the city in such a way that it would only last long enough for Earth to recover from the apocalyptic event. The builders speculate that the citizens of Ember should only remain in Ember "for at least two hundred years ... perhaps two hundred and twenty" ("The Instructions"). Therefore, the builders stocked the city with only enough provisions to last approximately two hundred years. The builders also left instructions for the citizens detailing the way out of Ember so that they can return to Earth's surface and begin repopulating Earth. However, since the builders did not want to create any panic or a premature mass exodus from Ember, they also kept the instructions a secret to be maintained by the mayor until the two hundred years had passed. Unfortunately, due to corruption, the instructions were lost, and the time to leave the city is now past due. Ember's power generator is failing, and the city is dangerously low on supplies; both of these problems put Ember in jeopardy of having its own apocalyptic event. Most of the characters in the story are willing to passively do nothing about the situation beyond hoard supplies for themselves and hope to be rescued. Lina and Doon, the protagonists, are the only two characters willing to actively pursue a rescue plan by searching for a way out of the city. As they search, they run into a conflict with the mayor, whom they discover is hoarding supplies for himself and not finding a solution to Ember's problem. They also run into a conflict with many members of society who are willing to keep waiting to be rescued rather than actively pursue solutions like Lina and Doon. Since Lina and Doon are at odds with their failing environment and at odds with the rest of society, we can clearly see that the central conflict in the story is character vs. environment.
But, as Lina and Doon pursue finding a way to save Ember, they battle with their own internal conflicts as well. Once Lina finds what is left of the important-looking paper that had been in the special box, she must battle being frustrated by others who don't see the document as important. Similarly, Doon must battle controlling feeling angry about the situation because, as his father says, anger can lead to "unintended consequences," and Doon must be more constructive if he is going to find a solution to the city's problem (Ch. 6). Both Doon and Lina must also battle with feelings of disappointment when they fail at first to find the door the instructions speak of and feelings of doubt that the instructions are important. All of these are character vs. self conflicts that Lina and Doon must overcome to bring the story to a successful resolution.