First we need to define the term. Atheism literally is a claim that there is no God. This is distinct from agnosticism which claims, "we don't know" when it comes to God. Sometimes the term anti-theism is used, which is like agnosticism but goes further to claim that all the evidence points away from the existence of God. Let's first deal with atheism proper.
Atheism is Irrational
The issue here is one of epistemology, the study of how we know things. In order to know there is no God one would have to have knowledge of his non-existence. As an analogy, if I claimed there are no aliens, I would have to have knowledge that there is no other intelligent life in the universe. Since I do not know this, and certainly cannot produce evidence that there is no alien life out there, it is not rational for me to claim, "there are no aliens." It may be my belief, but it would be an empty claim, based merely on my own experience rather than on reason. A lack of evidence or experience does not allow me to conclude that something does not exist.
The same is true of atheism. To claim, "there is no God," one would have to have complete knowledge that there is no God. How would one have knowledge of this? The best one could say is he or she has no personal experience or evidence of this God. But this does not lead to a rational claim of God's non-existence. I have no personal experience or evidence of aliens, but that does not mean there are none.
The problem is even worse than this for atheists. To know for certain there are no aliens, I would only need to have complete knowledge of the universe. This is far-fetched, but not nearly as far-fetched as the claim of atheism. God by definition is beyond the physical universe. To know for certain there is no God, one would need to have complete knowledge of metaphysics as well. In other words, an atheist would also need complete knowledge that there is nothing beyond this universe! The only one who could have such knowledge is...God Himself! Ironically, only an omniscient being could know for certain that there is no omniscient being.
To bring this out, when speaking to an atheist, simply ask for any evidence for the non-existence of God. They of course will not be able to provide any. To do so, they would need complete knowledge of the universe and knowledge that there are no metaphysical beings. This is why atheism is truly an irrational claim.
Agnosticism is Undefeatable, but also Unhelpful
Because of this, many would rather use the term "agnosticism," which again means "we don't know" particularly in relation to God's existence. The problem with this is it is truly an undefeatable position. That does not lend it credibility. We can claim ignorance on any number of things, which has no bearing whatsoever on its existence. By analogy once more, I could claim, "I don't know if dogs exist." Maybe I grew up in a part of the world where there are no dogs. I could be presented with pictures of numerous species of dogs, be shown taught various classifications of dogs, etc. and still claim I am an agnostic when it comes to dogs.
The reason for this is agnosticism is not technically an argument on God's existence. It is the lack of a position. How can you argue against the lack of an argument? What then does agnosticism add to the debate? Nothing, really. Back to the discussion on whether or not dogs exist. One tries to provide evidence of dogs to me, showing pictures and describing them. Another argues there are no dogs because they have no experience of them. A third person joins and says, I just want to add my position: "I don't know." Unhelpful.
Some might say, I don't believe in God because I lack evidence of his existence, but that really brings us to "anti-theism." One more word on agnosticism first. The existence or non-existence of God is of such importance that to remain in the agnostic position long-term is unwise. Much of the value and purpose of life, and the hope of eternity hinges on the question of God. Usually agnosticism means, "I don't know, so I will live as if there is no God." Practically then, it is equivalent to atheism.
Anti-Theism is Hopelessly Perspectival
That brings us to the third position: anti-theism. I can at least respect that this view tries to rationally come to a position. Again this is the view that says I cannot reasonably know that there is no God, but the evidence leads me to the position that he does not exist.
However, what evidence leads one to believe that there is no God? Back to the alien analogy. Could I have evidence against aliens? Yes...and no. If I could search the entire surface of Mars, I could claim there are no Martians, but I could not claim there are no aliens in vastly distant galaxies of the Universe. My claim would be based on my very limited experience and perspective. Having examined Mars, my claim that "there are no aliens in the universe" is still irrational, or at least overstretching.
How does this relate to anti-theism? Anti-theists argue that they have yet to find any convincing evidence for the existence of God. Keep in mind when speaking of God, we are speaking of a being vastly superior to us by definition (whether you believe in him or not), and who exists outside of the universe. Our search and study would be highly limited in determining his existence. Like the lack of Martians, no one is claiming that God only exists in a small sphere and should be easily recognizable. Quite the contrary, the claim is he is an infinite and eternal being beyond this universe.
There is more. What some see as clear evidence for his existence is dismissed by anti-theists. To name a few: The origin of the universe, the fine-tuning of the solar system and the Earth's environment in particular for life, the existence of morality, value, and aesthetics, and nearly culturally universal personal experiences with God. On that last one, it is true that these experiences differ, but this may be a clue to his vast nature. If I experience the Pacific Ocean a certain way from a certain shore, and 10,000 others experience it in different ways from different shores, we do not conclude the Pacific Ocean doesn't exist, but that it must be something vast indeed.
Keep in mind to argue for something's non-existence is far more difficult than to argue for its existence. Back to dogs. I have a dog named Teddy. I could argue for his existence very easily. All I have to do is show you positive evidence of his existence. If I argued that there are no other dogs in the universe (or in any metaphysical existence), it would be a far more difficult task. The anti-theist job is hard work indeed. He must dismiss all the evidence produced by those who hold there is a God, and hold that nothing beyond that evidence likely exists!
This to me smacks of not only arrogance, but really a hopelessly tribal and perspectival view. It is better than atheism, and more bold than agnosticism, but doesn't deliver much.
A Classic Red Herring
Many atheists, agnostics, and anti-theists will come up with all different types of objections to God based on such things as supernatural stories in the Bible, angels and demons, and peculiar religious customs. This is a red herring, it really just takes us off topic. These are questions about the nature of God, not the existence of God.
Let's take angels as an example. An atheist might argue, I simply can't believe that there are unseen spiritual beings all over the world at times intervening, thus I cannot believe in God. But God's existence is not at all contingent on the existence or non-existence of angels. It is exactly the opposite. First we must come to an understanding about God's existence, then the existence of angels follows. If God does not exist, then it is highly unlikely that angels exist. If God does exist, than the existence of angels is not hard to believe at all. It is part of the particulars that flow from a theist perspective. The same is true of the inspiration of the Bible and other religious customs; they only make sense in a world that first has God.
The most important example of this is the resurrection of Jesus from the dead. Many atheists, agnostics, and anti-theists argue this is irrational. But that depends first on whether God exists or not. If he does, then miracles such as the resurrection are completely rational. If not, then they are irrational. First things first.
Perhaps you are unconvinced of God's existence. I sincerely hope that will change. Knowledge of God's existence has provided me with purpose and a sense of value, a recognition of beauty and grace, and hope for eternity. However, if you are still seeking, for the meantime be careful of anyone who claims that atheism is the only rational position. It really is a faith-based belief that makes claims far beyond what is justified by reason and evidence.