The orthodox, textbook definition of 'atheism' is the belief that there is no God. This is the definition you'll find in dictionaries, encyclopedias, and in philosophical essays. For instance:

"Atheism is in fact extremely simple to define: it is the belief that there is no God or gods." -Julian Baggini

"Atheism is the belief that there is no God of any kind." -John Hick

"Atheism I take to be the denial of theism." -J.J.C. Smart

"An atheist disbelieves in the existence of God—he believes that there is no God. He doesn’t merely lack belief in a divinity; he positively believes in the absence of a divinity. Moreover, he takes his negative belief to be rational, to be backed by reasons. He doesn’t just find himself with a belief that there is no God; he comes to that belief by what he takes to be rational means—that is, he takes his belief to be justified." -Colin McGinn

"The traditional atheist believes that there is not now, nor ever was, a God." -Norman Geisler 

This has proven to be an unsatisfactory state of affairs for the new formulation of atheism, which rejects this heritage as meaningless piffle. It is customary nowadays to regard an atheist as one who does not believe in any gods and atheism to be just this disposition of unbelief. It is likewise customary to understand unbelief to be tantamount to disbelief. Thus deprived of any discrete descriptive referent, agnosticism is derided as either a wishy-washy emblem of ambiguity or an ideological refuge of cowardice for those afraid to commit to a position of perceived unpopularity or dogmatic extremism.

This interpretation is ostensibly abetted by a simple linguistic analysis. Since theism denotes the belief that there is a god, then it stands to reason, it is alleged, that atheism—which is taken to be the negation of theism—denotes the absence of such a belief. It is true that this interpretation has the advantage of etymological parsimony. However, the exigencies of semantic caprice and the strictures of logic do not favor this attempt to retroactively impose a spurious definition in defiance of centuries of philosophical tradition.

"If a person is designated as a theist, this tells us that he believes in a god, not why he believes. If a person is designated as an atheist, this tells us that he does not believe in a god, not why he does not believe," George Smith avers. And he is right in an trivial technical sense, but his aphorism is misleading. He concedes that a theist holds a positive belief—he believes that there is a god; but such a doxastic stance is altogether unintelligible unless he has at the same time reasons for his belief. A person is thus a theist only if there inheres in her declaration of belief this notion of 'why.' But an atheist, by Smith's reckoning, needn't account for his not believing in a god. Now we must agree with the latter part of his claim, because on this definition of atheism a person who, for instance, is conceptually deprived of whatever it is that could be a god should not of course be held to justify a non-existent proposition. But if a person is designated an atheist, this surely discloses more than the notion that he or she doesn't have a particular belief—for even if we hold Smith to the pointless designation of an atheist as any person just in case he or she is not a theist, the logical counterpart is nevertheless implicated by analogical balance in rejecting the theist's doxastic justification.

It is therefore plainly evident that whatever logical symmetry Smith means to convey with this argument is illusory. If we take atheism to represent the logical negation of theism we are left with an opaque appellation that does nothing to refine or clarify the concept of atheism. The set of belief-capable entities subsumed in this definition would broaden the register of genuine atheists to a point of inclusive inconvenience for those proudly self-designated atheists who, I trust, find little intellectual solidarity with imbeciles, infants, and farm animals. If a person expresses a positive belief in the existence of some other thing, a belief that this other thing exists, there may or may not be an actual ontological referent--the person could be wrong, after all—but there must necessarily be some explanatory relation of meaning. A theist has to qualify what it is that exists so that it should have to be represented by a finite set of predicates. It is hardly satisfactory to say that atheism represents the negation of theism, and thence to concede that the rightness of the one does not entail the wrongness of the other. If we suppose that atheists, in not believing in any gods, happen to be right, not only is theism left unrefuted but its claims are unconditionally immune to any of the tenets of atheism. This is hardly satisfactory if atheism is defined in the first place as a denial of theism.
   
Now were we to suppose instead that were no gods, then theism would be indeed be refuted; but this is an eventuality for which atheists should be entitled only to the satisfaction of not being explicitly wrong. This brand of atheism promises vacuous authority, however, because there are no grounds on which a doctrine which asserts nothing could be wrong.

Furthermore, there is no sense in taking the negation of theism or the denial of theism to signify the absence of a theistic belief. Theists believe that there is a god. The denial of this belief then is that there is no God. The claim that God exists is the salient claim of theism; to oppose this claim is to deny the content of the belief in God. It is the existence of the content or referent of the belief that suffers the negation, not the belief itself. If Jones believes that he can pass his exam, and Smith denies Jones's claim—that is, he says that Jones is wrong—we could not credibly infer that what Smith means is that Jones is wrong that he believes that he is able to pass the exam—that Jones does not in fact believe this. No, what Smith means is that Jones's belief is wrong; he does not have the ability to pass his exam. Now if we should stipulate that Jones does in fact believe that he can pass his exam—suppose, for instance, that believing such a thing is a necessary condition for being a Jones—the only coherent sense in which we could respond to Jones's belief negatively is to deny what he believes; such a denial necessarily entails the wrongness of its object. The evident grammatical polarity thus evinced is just a symptom of the logical duality of propositional denial.

This argument invites the rejoinder that the Greek prefix 'a' is alleged to convey merely a sense of detachment; that is, one who is an atheist is without theism. This point of pedantry doesn't respect the semantic encumbrance at issue, nor is not borne out by the conventions of the English language. One might say that such an interpretation is atypical. No, we must take the term 'atheism' to be an extensionally derivative modification of the term 'theism.' If it were to preserve the congenital naiveté alleged there should be no such appellation; the concept of atheism attains intelligible descriptive force only as a reactive judgment. Inasmuch as one can remain doxastically noncommittal only by incidental ignorance, the necessary conviction that should motivate allegiance to this category betrays a disposition that is necessarily incompatible with an assertion of unbelief.

Therefore whichever way the Greek prefix 'a' should be interpreted, its attachment to theism means only that agnosticism conceived in terms of doxastic futility and in terms of doxastic possibility do not advertise separate offers. It remains the case that you can know what the idea of god means and yet not commit to either option. As Mitchell Green points out, a satisfactory analysis of theistic epistemology "requires treatment of the omissive case. An assertion of ‘I do not believe p’ neither constitutes nor requires the assertion of not-p.' Goldstein holds that if I assert that I don’t believe that p then I deny that p. This is incorrect: an agnostic who truthfully reports, ‘I neither believe that God exists nor believe that He doesn’t’ would, on that principle, be making contradictory assertions about the existence of God. Surely that is not so, and proponents of this approach have not addressed the problem raised by the omissive case," he notes.

Furthermore any meaningful consideration of agnosticism has to countenance the simple fact that a disposition of agnosticism, however the term is defined, is compatible with not having a belief in the existence of God. It makes no sense, however, to conclude on this flimsy point of fact that any person fitting this description is an atheist just the same. Green makes this clear: "After all, someone who asserts ‘I don’t believe that God exists nor do I believe that He does not’ has not expressed the belief that God does not exist."

Now we have seen that the suppositional correctness of atheism in this model would fail to falsify theism. But even more damning to the argument is the fact that if God did exist, then atheism, under the definition that has become the vogue, would not be wrong. If an atheist were to thence countenance this contingency she would, of necessity, forfeit the title. Thus this disposition of atheism—in the putative absence of God's existence—should be preserved by doubt alone. But doubt, so considered, cannot meaningfully represent the vacuous epistemic condition of unbelief; rather, it is surely tantamount to disbelief because the negative formulation of atheism could meaningfully capture only a position of arrant conceptual ignorance and the people to whom such a description applies make fairly weak exponents of a doctrine they don't know anything about.

The analogical employment of Santa Claus (and his fictive counterparts) is a mainstay in the atheistic literature as a pedagogical prop that is supposed to clarify some relentlessly abstruse technical point about belief to show that agnosticism is really just atheism. But this simplistic thought-experiment serves to do nothing but conflate two very different doxastic attitudes. Admittedly, I take it that a person who doesn't believe in any Santa Clauses likewise believes that there are no Santa Clauses—or that, at least, there aren't any Santa Clauses fitting the fantastical claims of folklore. It is rather innocuous to equivocate disbelief and unbelief in this case, for the existence of any such entity is amenable to empirical disconfirmation. The idea of Santa Claus is not bound up in the metaphysical; it would be odd indeed to make or to dispute the claim that his existence is logically necessary. This hackneyed argument is an analogically vacuous imposture, and a symptom of the fatuous tenets of new atheism.