On a practical level, apatheism eschews positively asserting the existence or nonexistence of gods. Apatheism must therefore be regarded as an attitude towards a type of belief, not a belief or disbelief itself.
Apatheism sometimes goes a bit further and asserts that even if it were proven conclusively and without a doubt that some sort of god existed, then the person's general behavior and life would not change — thus the existence of gods is not only irrelevant now but would be irrelevant in the future no matter what sort of evidence or proof is present.
In some cases, apatheism is applied more broadly to all religions and even to all belief systems and ideologies, not just to belief and disbelief in the existence of gods. This broader sort of apathy and apatheism would be more correctly labeled Indifferentism, though because that label comes from Catholic theology it's not one familiar to most people.
There is some question of whether apatheism should also be applied to the beliefs of others. It's one thing to be apathetic about belief and disbelief in gods for one's self, but quite another thing to be apathetic about others' belief and disbelief in gods. It would be difficult to defend the proposition that others' belief and disbelief in gods don't affect you, thus it would be difficult to justify being apathetic about that.
Apatheism, Theism, and Atheism
Since any given belief is either present or not with any given person, even apatheism is not truly a "third way" between theism and atheism. Apatheism is apathy towards both the existence of gods and belief in the existence of gods, but apathy towards something doesn't mean that it's absent. Thus apathy towards belief and disbelief in gods doesn't exclude belief and disbelief in gods.
So apatheism is compatible with both atheism and theism, which means that apatheists are also either atheists or theists. An apatheist who is also an atheist doesn't believe in any gods and also doesn't regard that disbelief as important for their lives. That can be reasonable because there are lots of things that we all disbelieve in but also don't care about — who disbelieves in elves and also regards that disbelief as important in their lives?
An apatheist who is also a theist has some sort of belief in the existence of some sort of god but doesn't really see the existence of that god as important in how they live. This can also be reasonable because we all believe lots of things that, as a practical matter, aren't really important in our lives.
Apatheism & Morality
In the past apathy towards gods and the existence of gods was treated as a moral failing, but in recent years apatheism has been received more positively. Jonathan Rauch wrote in the The Atlantic Monthly in May, 2003:
Apatheism is to be celebrated as nothing less than a major civilizational advance. Religion, as countless acts of violence in the name of God have underscored, remains the most divisive and volatile of social forces... Apatheism, therefore, should not be assumed to represent a lazy recumbency... Just the opposite: it is the product of a determined cultural effort to discipline the religious mindset, and often of an equally determined personal effort to master the spiritual passions. It is not a lapse. It is an achievement.
It seems likely that while the term apatheism is not widely known, actual apatheism is quite common. In America around 40% of adults regularly claim in polls that they go to church regularly. When church attendance is studied directly, though, the best estimates find that only around 20% of adults really do attend church regularly. Thus Rauch concludes:
A great many Americans ... apparently care about religion enough to say they are religious, but not enough to go to church.
Apatheism may be defended in a couple of different ways. What might be called a more implicit argument for apatheism is that the supposed benefits of theistic religion, like psychological comfort and morality, can be found at least as well if not better in secular sources thus theism and religion simply aren't necessary anymore.
A more explicit defense of apatheism would be that theistic religion tends to cause more harm than good, especially when it comes to division and rancor in a society. More widespread apathy towards religious theism would entail apathy towards religious distinctions, religious differences, and religious dogmas, which in the long run would result in fewer division based on religion.
Apathetic theism can be defended with the argument that God's failure to provide obvious signs like miracles indicates that he doesn't much care if humans believe in him. Therefore, apatheism is a reasonable position unless and until God shows greater interest in what people believe and demonstrates that such belief should matter to us.