The Believers are a group of Ember's citizens who get together and sing happy songs about what they, you know, believe in. They can be seen around town, doing their thing: "There a stringy-haired man sat cross-legged on the ground playing a flute made out of a drainpipe, and five or six Believers circled him, clapping and singing" (5.20). Seems cheery—if a bit cult-like.
What do they believe in? Captain Fleery, who's a Believer, explains to Lina that help is coming: "the Builders will come again and show us the way" (7.32). Why do they think this? Well, Captain Fleery says that she knows it in her heart, "And I have seen it in a dream" (7.36). Sounds like these people aren't really into evidence-based faith.
Sometimes they just seem downright irritating, too, like when Lina and Doon are trying to meet in Harken Square to discuss Serious Business: "At the north end of Harken Square stood a circle of Believers, clapping their hands and singing one of their songs. Lately they seemed to be singing more loudly and cheerfully than ever. Their voices were shrill" (12.26). Can we get some earplugs please?
But in any case, the Believers also represent the dangers of inaction. When everything really hits the fan, we readers know that the Builders aren't too likely to show up. It's folks like Lina and Doon who will save the day, because they believe in themselves enough to take action and get things done.