Morality tries to answer the question, "why shouldn't I harm others if I want to?" There are several reasons people can give for why you shouldn't initiate force or fraud on others. Let's look, as an example, at how I might convince you not to steal:
1) Enlightened Self-interest - If I tell you not to steal because in the long run it destroys that which makes a thriving civilization possible, then I'm giving you a long-term, indirectly selfish reason not to steal.
2a) Eternal Damnation - If I tell you not to steal because this life is really just an audition for a second, infinitely long life that begins when this one ends - and there's this all-powerful God who will punish you forever, then I'm also giving you a long-term, directly selfish reason not to steal.
2b) Karma - If I tell you not to steal because you're going to be reincarnated after this life ends, and your status in the next life depends on your behavior in this life, then I'm also giving you a long-term, directly selfish reason not to steal.
3) Empathy - If I tell you not to steal because if you put yourself in the victim's shoes, wouldn't that feel terrible, then I'm giving you an immediate, directly selfish reason not to steal. This argument depends on you being able to feel empathy. (This is an effective tool for perhaps 98% of the population, and is one of the best tools for teaching morality to children. I think it's so effective because reason #1 is so compelling that it was selected for by evolution.)
4) Crime Does Not Pay - If I tell you not to steal because I & the government I support will getcha if you do, then that is an immediate, directly selfish reason not to steal (or at least to make sure you don't get caught).
5) Duty - If I tell you not to steal because "it's just wrong", then I'm invoking a reason that does not appeal to your selfishness at all. In fact, this argument doesn't appeal to any facts about the real world - it's essentially an arbitrary statement! This sometimes works with children if they recognize me as an authority figure. In that case they implicitly trust that I have a good reason to say it's wrong but for some reason I don't want to explain it to them. But adults demand explanations, so if I use this argument on an adult it basically amounts to me begging you to please please don't steal.
Notice that there is only one argument for moral behavior that does not ultimately appeal to your own self-interest, and that one is arbitrary! This is a pity, because at first blush #5 sounds like it should be the most compelling argument of all: It sounds so final & absolute.
Theists' moral systems are based on #2a, and sometimes #5. Usually they will acknowledge the truth of #1, though they don't think that's enough of a reason to be persuasive. In practice their system would have to depend on #2a, 3, & 4. Atheists' moral system is based on #1. We think that #2a & 2b are factually untrue or (unprovable & therefore moot). In practice our system depends on #1, 3, & 4.
Also notice that atheism & theism both have fundamental rationales for moral behavior. In practice both atheists and theists can agree that #3 & 4 must be supported by a healthy society.
Without morals a republic cannot subsist any length of time; they therefore who are decrying the Christian religion, whose morality is so sublime & pure, [and] which denounces against the wicked eternal misery, and [which] insured to the good eternal happiness, are undermining the solid foundation of morals, the best security for the duration of free governments. (Source: Bernard C. Steiner, The Life and Correspondence of James McHenry (Cleveland: The Burrows Brothers, 1907), p. 475. In a letter from Charles Carroll to James McHenry of November 4, 1800.)
[O]nly a virtuous people are capable of freedom. As nations become corrupt and vicious, they have more need of masters. Source: Benjamin Franklin, The Writings of Benjamin Franklin, Jared Sparks, editor (Boston: Tappan, Whittemore and Mason, 1840), Vol. X, p. 297, April 17, 1787.
Of all the dispositions and habits which lead to political prosperity, religion and morality are indispensable supports. In vain would that man claim the tribute of patriotism who should labor to subvert these great pillars of human happiness, these firmest props of the duties of man and citizens. The mere politician, equally with the pious man, ought to respect and to cherish them. A volume could not trace all their connexions with private and public felicity. Let it simply be asked, Where is the security for property, for reputation, for life, if the sense of religious obligation desert the oaths, which are the instruments of investigation in Courts of Justice?
And let us with caution indulge the supposition that morality can be maintained without religion. Whatever may be conceded to the influence of refined education on minds of peculiar structure, reason and experience both forbid us to expect that national morality can prevail in exclusion of religious principle. It is substantially true, that virtue or morality is a necessary spring of popular government. The rule, indeed, extends with more or less force to every species of free government. Who, that is a sincere friend to it, can look with indifference upon attempts to shake the foundation of the fabric? (Source: George Washington, Address of George Washington, President of the United States . . . Preparatory to His Declination (Baltimore: George and Henry S. Keatinge), pp. 22-23. In his Farewell Address to the United States in 1796.)