When we think of a mountain, we necessarily think of a valley — they are inseparable concepts. Similarly, Descartes argues, when we think of God as a supremely perfect being, we necessarily think of God as existing. Existence is itself a perfection, and a being lacking it would not be supremely perfect. Therefore God must exist.
The obvious reply: we cannot think a thing into existence. Descartes anticipates this. The mountain-valley analogy shows that the necessity comes from the nature of the things, not from our thought imposing it. What distinguishes the God case from a perfect island — or any other invented perfection — is that God's essence uniquely requires existence in a way a perfect island's does not.
The argument is not merely a proof of God. It is structurally essential to Descartes's project. Once God's existence is certain, God's non-deceptive nature follows. And once God's non-deception is established, the whole guarantee of clear and distinct ideas is in place — and the edifice of knowledge can be rebuilt on secure foundations.
Kant's is the most famous objection: existence is not a predicate. Adding 'exists' to the concept of God adds nothing to the concept — it only asserts that the concept is instantiated. The argument was first put in this form by Anselm in the eleventh century. Contemporary philosophers including Plantinga and Malcolm have defended modal versions of it.