It often happens that when someone mentions the word "philosophy", another person will mention the tree that fell down in the forest. They're referring to the question: if a tree fell down in a forest, and no-one heard it fall, did it really fall? At first, it appears as a silly question. Of course it fell! But it's actually quite a serious problem that was pointed out in Bishop Berkeley's argument. Another common misconception that people have on philosophy is that is heavily concerned with spiritual matters, vague and meaningful. This is heavily misguided; philosophy is the rigorous pursuit of truth. It is not vague, and mysterious; it is clear, and logical. However, the arguments concerning God do form a branch on the tree of philosophy.
Why do we need to question the existence of God? Because we're rational, and we really want to make sure he's there. So we use the best methods we have of testing truth. In science, it's using the senses. In philosophy, it's logic. If God existed, we should either be able to see him, or come up with a good philosophical argument.
Definitions are important in philosophy. To understand what something is, we need to know some things about it. Here is the standard definition of the Christian God: he is omnipotent (absolutely powerful), omni-benevolent (absolutely good), and imniscient (knows everything).
A problem with this definition was found by medieval scholars, who grew very concerned over its conclusion. The problem came from the question: could God cease his own existence? Note that the question is not concerned with whether he would, but whether he could. According the definition, God is omnipotent. He can do anything he wants. Ruling out the ability to kill himself would infringe on his almighty power. So God must be able to commit suicide.
What worried the medieval scholars was that if God were to kill himself, we wouldn't even know.