What does it mean to exist for Descartes? Of course I know about cogito ergo sum, but is that all there is to it? Is the only criterion to exist to be able to think? Or is there more to it? And if it is the only one, what kinds of consequences does that have?
I don't think Descartes has a special idea of what it means for something to exist.
I think you may be confused when you write:
Is the only criterion to exist to be able to think?
Descartes' argument in Meditation 2 is quite different from what a simple reading of "I think, therefore I am" would mean.
Specifically, we need to grasp a few things that have already happened in the meditations:
- In Meditation 1, Descartes talks about often being deceived -- especially by his senses.
- He then decides to embark on a quest to find certainty by abandoning everything uncertain and clinging to only what is certain.
- He flirts briefly with the idea that he's being manipulated but abandons it -- in part because if he accepts he cannot get off the ground at all.
Thus, he's starting from a place where he doubts:
- His senses and everything that comes from him
- The content of his thoughts
What he accepts:
- There's no deceiver jamming up his thinking
- A certain set of reasoning skills
From this, he constructs an argument that looks at this phenomenon:
He's experiencing thoughts.
He proposes two possibilities:
- He is experiencing true thoughts (what he thinks is true)
- He is not experiencing true thoughts (what he thinks is false)
One of the thought is:
If possibility 1, then Descartes exists. The inference here is if his thoughts are true and one thought is "I exist", then this thought is true. Therefore, He exists.
If possibility 2, then Descartes also exists. The added premise here is that only things that exist can think and be deceived. Thus, if he has a false thought, then he is deceived. If he is deceived, then he exists.
Thus, by constructive dilemma (take either fork and get to the same place), Descartes infers he exists.
This doesn't amount to a blanket "if someone else has thoughts, I know they exist."
To return more generally to your question about Descartes' views on existence. There's no where in the Meditations where Descartes draws a generic connection between existing and thinking. He does draw a specific connection between his having thoughts and his existing -- but this argument does not extend to others. Moreover, based on later Meditations, it's clear Descartes thinks lots of things exist including things that are non-thinking -- like rocks, towers, candles.
Descartes did not explicitly discuss the meaning of existence, i.e. ontology. That is why Heidegger says about him things like
With regard to the ontological development of the problem, Descartes is far behind the Scholastics. He actually evades the question. (Being and Time §20)
Descartes's implied concept of existence is mostly standard Aristotelian:
An attribution of an intrinsic property implies existence. E.g. "I think" implies "I exist".
A cause of an existent is an existent. This underlies two of Descartes's proofs for God's existence. E.g. the cause of my idea of infinity must be an actual infinite being.
Existence in a primary sense is attributed to "substances", that is to beings that "stand on their own". Descartes's interpretation the substance idea differs somewhat from Aristotle's. Descartes concludes that the entire material world is one substance. Individual minds are also substances, and also God.
In addition, Descartes's use of the Ontological Argument concerning God's existence implies the following "non standard" views about existence:
That existence is a kind of quality, and moreover a kind of perfection. Since God is the most perfect, he must possess existence, which is one of the perfections.
That everything that is thinkable is possessing a sort of existence (*) just by being thought about. So God exists, firstly, simply because we can think about him. And, secondly, since God is thought about as the most perfect being, he must possess normal existence (**) too, because normal existence is a perfection (4).
(*) objective reality in Descartes
(**) formal reality in Descartes
God necessarily exists because of how Descartes reads idea-causation and what the content of the idea God hence ontologically implies (and is reliant upon). God is the only Cartesian substance qua Cartesian substance, strictly speaking, since it is the only thing that can exist on its own. But the body and the mind are each substances in a looser way because Descartes gets a clear and distinct perception of them such that he can conceive of them being able to exist independently from one another and on their own (though relying on God for persistence). I don't see why in principle Descartes would reject abstract objects, but he made it seem like we didn't need them, or that God would just give us what we need rather than make a separate realm(s) for things like propositions, sets, mathematical objects, etc.
Descartes regards his cogito ergo sum argument as a premise (i.e. an assumption), rather than a proof or a conclusion. Hence, to Descartes, for an object, the quality of being conceived is strictly less general than the quality of existing. Thus, to Descartes, there might be unconceivable objects which exist, but there are no conceivable objects which do not exist.