What happens to non-believers when they die? How about if they're generally "good people" (as in they have good hearts)?
Let me answer the 2nd question first. Since the dawn of time, human beings have tried to justify their lives (and earn their way to heaven/God) in one of three ways:
1. They assume they are living a good enough life to please God when they finally meet Him.
2. They work very hard at some social cause because they think that they are good people and need to overcome the evil deeds of bad people
3. They work very hard at keeping the rules of a particular religion in order to justify themselves as a good person in the sight of God*
While each of these "good people go to heaven" options seem reasonable, this theory does not make sense and is full of problems (read this post from last week for a full explanation, "Why isn't being good enough sufficient?").
Scripture tells that the biggest problem of all is that there are no "good people" out there. All human beings are sinful (Romans 3), meaning we are all bad people. In the Gospel of Luke (18:19), Jesus actually states that the only person who is good (perfect, pure, sinless) is God himself. Human beings are imperfect, impure and sinful. And the reason is because we do not have "good hearts." In fact, all of our sin comes from the very core of our being, or what the Bible calls the "heart" (Mark 7:21-23). Out of the sinful heart comes sinful thoughts, words, deeds and intentions.
This leads us to your second question: What happens to non-believers after they die? Simply put: They are condemned before God for their sin. They are subsequently sent AWAY from Him, which is a frightening thought because God is the source of all beauty, goodness and love. So when unbelievers are removed from His presence forever, they enter into a place absent of all beauty, goodness and love. Scripture calls this place Hades or Hell. Now this sounds harsh, but consider what C.S. Lewis said: "Hell is the greatest monument to human freedom. All in hell choose it." Lewis is saying that those who reject God's salvation in Christ freely do so. They choose NOT to be with God, so God respects that choice.
Friend, God is giving you a choice today. I implore you to choose God by choosing to trust in Jesus for you salvation and NOT your own works/heart/deeds/goodness. Christianity is different from all other religions because it says that we can't do anything to earn our salvation. The standards are too high; and we are too weak and sinful. The only way to be saved, therefore, is by trusting the only "good person" that ever lived: Jesus. Jesus lived the perfect life we could not live. He died the death that we should have died. In doing so, Jesus takes away the sin of those who trust in Him and grants them access to God, i.e. salvation.
[Answered by Pastor HM]
*Mark Driscoll, Religion Saves and Nine Other Misconceptions (Wheaton: Crossway Books, 2009), 159-160.