i have just written lemmie see....four essays in my AS philosophy exam on proof for an against God, believing in God is not like believing in santa clause. Arguments give many reasons for why god could exist. The ontological argument-"God is that than which nothing greater can be concieved, but you are also something greater than can be concieved. Indeed, since it is possible to be concieved to be something of this kind, if you are not this very thing, something can be concieved greater than you, which cannot be done", St Anselm, Proslogion 2. He exists in the mind as each and every person can come up with some way to describe him so each of us has grasped the concept of god whether we believe or not. God is necessary (apparently) as something that can be thought not to exist would be inferior to that wich cannot exist altogether, thus Anselm has made a demonstration though ok it' not a great one. Descartes said that god must exist because existance is a predicate of a perfect being, therefore he has to exist to avoid being self-contradictory. To say God has no predicate of existance is like saying you can have a triangle with no sides-it's illogical.
Cosmological argument-look at everything around you, everything including you. Everything was caused nothing simply comes into existence onit's own accord.Everything has come into existence because something caused them to. the causes can go back to the beginning of time, time began with the creation of the universe, there must have been something to cause the universe, it would have to have necessary existence to cause such a contingent universe. God has to have necessary existence and so is the first cause. Aquinus had a ton to say on this.
Teleological argument
the universe shows purpose an regularitym the complexity of it shows design, design implies a designer i.e God or some other supreme being. Paley had a bash at this one likening the world to a watch, if you're not a skilled watchmaker and you put a watch together it probably won't work-you are not a watch designer or maker. Give it to its designer they'll have it done in no time. So in order the regularity an purpose of the universe to take place a skilled overseer must have put it together that way.
There's the moral argument about how each of us strives to goodness to achieve the summum bonum or greatest good. the greatest good must be a fair good an paradise that we achieve when we have fullfilled our moral duty through kant's catagorical Imperitive thus giving the idea of heaven. heaven is a christian concept so the existence of god is a presupposed concept. that argument is actually alot more complicated!
Good ol religious experience so mental, spiritual and paranormal experiences where a supreme being is shown or noticed or knowledge of one is given. Lotsa people have them and suddenly believe that this supreme good or being must be God (or a higher being)
there's also your basic paranormal stigmata, statues bleeding or crying and the like.
I myself don't believe in God but to say it is impossible for him to exist is plain daft. There are huge amounts of evidence for and against an more often that not the for weighs out the against.