Ideas must first be coherent before they can be scrutinised. If I were to ask "do you believe in Quixido?" would it make any sense to comment on the ontological status of Quixido without knowing what Quixido is? When it comes to God we do at least have a fair idea of what the conception of God is. While many would disagree over God's nature, there's at least a general sense by which the concept can be understood. Far from Quixido, God involves a consciousness and intelligence whereby the universe in some capacity represents an expression of that intelligent conscious will, and generally speaking it has some interest in the affairs of humanity.
So an atheist in the very broadest is someone who rejects that notion, it's a descriptor for those who do not think such an entity is a valid one. This is to be distinguished with noncognitivists who have no idea of the concept, and strong agnostics who take such an entity as being unknowable. For myself I tend to fluctuate between atheism and agnosticism depending on the specifics put forward because depending on how God is conceived determines its coherence.
Is God possible?
When it comes to the conception of God as being omnipotent, omniscient, and omnibenevolent, the existence of such a concept is seen to be incompatible with the existence of evil. Such a conception of God can be ruled out, and indeed much time and effort has gone into addressing this concern. The problem of evil is justification for being a strong atheist, at least in respect to conceptions of God that involved absolute power and absolute goodness.
Likewise other traits are problematic. To be both omnipotent and omniscient brings contradictions (can God act in such a way that even he won't know about?), as does the trait of omnipotence on its own (can God make a rock so heavy that he can't lift?). These kind of proofs though are at least to me unsatisfying because while they ground God in definition they don't capture the essence of what God is. Someone praying for God to cure their cancer doesn't concern themselves with the paradox of omnipotence, but curing cancer irrespective of limits to such traits is something God is meant to be able to do.
Then there's the problem of the supernatural. Can something supernatural act within the natural world? If something is acting in the natural world what stops it being natural itself? Can something simultaneously be in and outside of time? Can something outside of time experience or be said to have a thought? Such problems would make some conceptions of God impossible too, again grounds for strong atheism.
Is God plausible?
While some conceptions of God can be ruled out a priori and thus we have justification for strong atheism, this doesn't cover the range of different conceptions that could be reasonably called God. Perhaps while not impossible, there's good reason as to why they aren't there.
A weak atheist could state that there's no good evidence for God. While absence of evidence doesn't make evidence of absence, the lack of any good instances of observed intervention count against an interventionist deity. An undetected and undetectable deity is indistinguishable from there being no deity at all. Claims are often made, sometimes miraculous in nature, but these claims have to be put into context of those who claim alien abduction and those who fake bleeding statues and the fallibility of human memory. The reliability of the testimony has to be weighed against the possibility of false testimony. This is the problem of miracles.
A weak atheist could also state the problem of design. While it is said design requires a designer, any intelligent design we know (a pocket-watch for example) comes not ex nihilo but from a complicated designer. The ability to make a watch is something that has evolution both biologically and culturally to allow for such design. "Who designed the designer?" isn't just a witty retort, it's highlighting that a designer doesn't solve the problem of complexity that its being used to explain in the first place.
A weak atheist could also state the problem of physical minds. As far as empirical inquiry into the nature of the world, the evidence clearly points towards mind being an emergent property of physical forces. Altering the brain alters conscious experience. Damaging the brain affects cognition. This leads to problems for the nature of God as well as notions like a soul or an afterlife that are tightly-coupled with some concepts of God.
A weak atheist could also state the burden of proof. While an atheist cannot disprove God they cannot disprove Thor or Ra. Nor could they disprove Santa for that matter, or an alleged china teapot orbiting between Earth and Mars too small to be detected by any instrument. The burden of proof is on the one making the existential claim, because there's an infinite number of things that cannot be demonstrated. A weak atheist would be justified in stating the absence of a good case for God is a good case for atheism.
Is God necessary?
Despite problems associated with concepts of God, there are those who say that it doesn't matter about the problems associated with God or how unlikely it is because God is necessary.
The most common of these is the first cause argument, that natural causes cannot account for existence itself. The notion of a prime mover went out the window with the notion of a clockwork universe, with notions like the arrow of time and causality probabilistic rather than definite. And even if the argument did hold it is at best an argument for deism, but the question of if a natural cause requires a supernatural one, does a supernatural cause require a supersupernatural one?
While the teleological argument is in occasional use for life itself, it is commonly found today in the cosmological constants. In essence, the universe is fine-tuned for life because if some of the variables are off by only an a fraction then life as we know it wouldn't exist. While whether there are any fine-tuned variables is a debate for physicists there's a more fundamental problem with this reasoning. There's no reason to assume that life, and more specifically us, is the focal point for the laws of physics. Evolutionarily so much owed to our existence is a product of contingency that it makes no sense to privilege us more than any other life-form.
The argument from morality is another necessity that could be said to be only the product of a divine being. The most basic objection to this is that evidentially we can see degrees of what we could consider morality in other animals. Morality, evidentially, clearly is an evolved trait and requires no further explanation.
The transcendental argument argues that for absolutes there needs to be an absolute mind. 2+2=4 because God holds it to be true is one such transcendental argument. But this argument holds 2+2=4 could just as easily be true as 2+2=231214 2+2=white. It makes absolutes arbitrary, and if by their nature universal then why bother posit a deity to explain it in the first place?
Is God explainable?
While there are many things wrong with the concept of God, there's something to be said for the cultural and psychological factors that enable belief in the first place.
The first challenge to God is that it's just the construct of our culture. The reason that we're asked about the Christian conception is that we've grown up in a predominantly Christian society. Instead of being asked about the historicity of Muhammad ascending to heaven on a winged horse, we are told about the historicity of Jesus conquering death. No dining in Valhalla for us in the modern western world or breaking the cycle of Samsara, just eternal punishment or reward because of the sacrifice of the Godman on the cross. Furthermore that construction has changed throughout history, the Old Testament has God fighting battles while these days God shows Himself by burning the image of Mary into a grilled cheese sandwich.
But what accounts for the belief itself? We have evolved certain ways of thinking. We are intuitive physicists, intuitive biologists and intuitive psychologists. These processes are important as they allow us to distinguish between different interactions. We are wired for agency and in particular human agency. Even seeing design is something intuitive for a lot of people.
In modern times we have seen the birth of religions, Mormonism is less than a couple of centuries old while Scientology is approaching its 60th birthday. Then there's the New Age movement where not only are there claims of extraordinary powers but personal belief in possessing them. Alien abductions, conspiracy theories like 9/11 being an inside job or the staging of a moon landing, these are part of our culture despite their respective implausibility. New beliefs in crazy unsupportable things arise all the time and some even gather significant support even though there's no good reason for it to do so. In this respect, explaining God belief is just one of the many weird things that permeates in our species without good reason.
Of course being able to explain the cultural phenomenon doesn't mean the phenomenon doesn't exist, but what it does do is take away the sense that the fact that it's so pervasive in and of itself dictates further insight.
The position of atheism is justified for a number of reasons. That there are no good reasons for God is sufficient grounds for weak atheism, and that there are good reasons against thinking that such a construct is likely if it's even possible. Arguments for the necessity of God don't make such a case and sometimes are downright absurd, and the case is made even worse as God is perfectly explainable as a personal and cultural construct without any need for there actually being such an entity. The case for atheism is very solid, and should give any theist pause for thought. The notion that we should all be agnostic in the weak sense seeks to neglect what is already known on the subject, taking one's personal ignorance and projecting it onto people at large.