Objective, atheistic morality begins with the axiom of existence. In other words: I exist. From this axiom, we can ascertain the value of existence, or the value of life. In fact, the value of one's own existence is just as objectively true as the fact of one's own existence. This is because it is impossible to conclude that one's existence as bad or undesirable without first conceding that it is indeed valuable and precious. If someone says, "It is bad that I exist, and I should not exist," they are making a big mistake, for in their mere act of concluding such a thing, they are presupposing the value of their own existence. This is because one cannot make such a conclusion without existing in the first place, and if one were to not exist, it would be impossible for them to conclude that it is good that they don't exist! Human morality cannot be used to conclude that humans should not exist, for without humans, there would be no human morality to use to reach that conclusion.
With me so far? Good. Now an objective, atheistic morality says that what is moral is what is good for you. It also says that what is good for one human is good for all humans, since all humans exist in the same reality and all the rules apply universally. In fact, if a rule cannot be applied universally, then it is not really a rule. So the only moral "rules" that exist are the rules that apply universally.
From here, the objective, atheistic morality says that all moral justifications have to be based on one's own self-interest. I presuppose the value of my existence, I operate under the same moral rules as everyone else, and my moral choices involve me choosing the action that best serves my self-interest. Self-interest is a powerful concept, and is indeed essential to morality. Through self-interest and universality, I can say that it is wrong for me to coerce someone because it is wrong for me to be coerced. If I commit a coercive action against someone, then I am implicitly stating that it is ok for me to be coerced. I would be violating my own self-interest, and therefore would be acting immorally.
This moral code is objective because it looks at the facts of reality and one's own existence. This moral code is atheistic because it requires no God, requires no afterlife, and requires no immaterial entities or dimensions for its use.
Theistic morality, on the other hand, says that you should obey God because either he tells you to, or you will roast in Hell, or a combination thereof. In my appearance on The Atheist Hour, Pastor Gene Cook said that we should do what God says because He says so and because we should be devoted to Him and obedient to Him. When I pressed Gene further, he admitted that Hell is a punishment for those who fail to follow God's word. But then I took it a little bit further. You see, the Christian morality does not admit the concept of self-interest; it does not recognize the axiom of existence and how we conclude one's inherent worth in their own existence. The Christian moral code steals from the objective, atheistic moral code. It steals the concept of self-interest, and it steals the concept of the inherent value of one's own existence.
In other words, all Godly/theistic moral systems presuppose the objective self-interest based morality as a foundation, and their theistic moral systems are nothing more than accessories, despite the fact that they masquerade as being foundational.
Here is an example dialogue where anyone can expose the theistic moral problem I described above. I managed to quickly sneak it in on Gene Cook's show, but he didn't want to dwell on it. I suggest you try it with your own local Christian and see what results you get!
Atheist: What's your morality based on?
Christian: God and His holy word. What's yours based on?
Atheist: My own self-interest. In fact, so is yours. You just fail to recognize it. May I ask for what reason would you follow God's holy word?
Christian: Because he says so and because you have to love Him.
Atheist: But why should I care what God says or follow his commands? What will happen?
Christian: He will forgive pretty much all your sins if you believe in Him, but you will be cast into Hell if you don’t believe and submit to Him.
Atheist: So your moral code is a carrot/stick approach, huh? Sounds like coercion to me. But you still haven’t given me an "ought," Christian. Why should I care if I get sent into Hell and burned forever and cast away from God?
Christian: Because pain hurts and nobody wants to be hurt or destroyed.
Atheist: But why? Why should I care if God hurts me or destroys me and I feel pain because of Him?
Christian: Ummmm... because God tells you to?
Atheist: And where does the Bible say that you should presuppose the value of your own existence and avoid pain and destruction? You see, you have no reason to follow God's word, or care what he does to you, without the foundational objective morality that I have presented. The only reason you follow God's word is because you presuppose the value of your own existence and happiness. The value of your own existence and happiness is the foundation of your moral system, and your Christian morality is nothing more than an accessory that is wholly dependent upon the foundation of your own self-interest for it to have any meaning.
Christians sometimes claim that atheistic morality steals from theistic morality. This is simply not true. I've just demonstrated that theistic morality is not even a moral system at all because it is not foundational; it is an accessory to the foundational moral code of self-interest from which it steals.
Morality is about each of us making the best choices for our lives. Morality is not about wholly submitting to another conscious entity a priori, regardless of how omnipotent or omni-whatever that entity is. And the only way you can even justify wholly submitting to another entity is by taking it to the foundation and showing that this submission would be the best choice for one's life. But either way, morality can only be based on principle(s), not specific commands or third-party entities. Morality is founded upon the principle of rational self-interest, not on some creator God, nor his dictates.