Apatheism is acting with apathy, disregard, or lack of interest towards belief, or lack of belief in a deity. An apatheist is someone who is not interested in accepting or denying any claims that gods exist or do not exist. In other words, an apatheist is someone who considers the question of the existence of gods as neither meaningful nor relevant to his or her life; nor to human affairs.
Personally, I see Ignosticism as an acceptable position to take regarding the question of the existence of god. After all, each religion defines god in fundamentally different ways. Even the three main "Abrahamic" religions, which I've often been told worship the same deity, have crucial differences in their definitions and claims over who or what this deity actually is.
Apatheism however, as it is defined above, I do not accept as an intellectually tenable position, though I know many individuals who would seem to fit the definition perfectly. Assuming one could avoid an ignostic response by first defining the word "god" (let's assume it's an omniscient, omnipotent being that created the universe and takes an interest in our activities), then the correct answer to the question "Does god exist?" is crucial, and would have an effect on our lives that would be absolutely profound.
If the answer was yes, further questions would immediately arise, most notably "So which religion is the correct one?" and "How can I avoid upsetting such a powerful being?". Given the existence of god, these questions would be of fundamental importance and would utterly dominate our lives (which begs the question "do believers really believe?").
If the answer was no, think how drastically the world as we know it would change, for better or for worse (I would argue for the better). So many aspects of society (and therefore our lives) are built upon the foundations of religion, and to ask "is there a god?" shoots straight to the core of whether or not these aspects are justified.
The existence of god is therefore a vital matter, and, even if the question may never be resolved, to take the apatheist's stance is to admit to indifference to the world and intellectual lethargy.