Probably the best (or at least a very popular) argument for God's existence is the Argument from Design, or Teleological Argument.
There have been a number of proponents of these arguments, from St. Thomas Aquinas, who argued for God's existence in his Summa Theologica to David Hume the noted empiricist and philosopher in his Dialogues Concerning Natural Religion.
In summary, the argument (in its many forms) states that there is order in the universe, and because of this, the universe must have been designed. Therefore, God exists. This is a gross oversimplification, but it's pretty close to the mark. Atheists and philosophers have been dissecting and debunking these arguments for centuries.