1. What is the mark of the mental? Brentano took intentionality to be the mark of the mental, the criterion whereby physical and mental phenomena are distinguished. For Brentano, (i) all mental phenomena are intentional, (ii) all intentional phenomena are mental, and (iii) no mental phenomenon is physical. (Franz Brentano, Psychologie vom empirischen Standpunkt (1874), Bk. II, Ch. 1.)
2. What is intentionality? ‘Intentionality’ is Brentano's term (borrowed from the Medievals) for that property of mental states whereby they are (non-derivatively) of, or about, or directed to, an object. The state of perceiving, for example is necessarily object-directed. One cannot just perceive; if one perceives, then one perceives something. The same goes for intending (in the narrow sense), believing, imagining, recollecting, wishing, willing, desiring, loving, hating, judging, knowing, etc. Such mental states refer beyond themselves to objects that may or may not exist. Reference to an object is thus an intrinsic feature of mental states and not a feature they have in virtue of a relation to an existing object. This is why Brentano speaks of the "intentional in-existence of an object." Mental states for Brentano are object-directed by their very nature as mental states: there is no need that a particular state's object actually exist for that state to be intentional. It follows that intentionality is not, strictly speaking, a relation. For, necessarily, if a relation obtains, then all its relata exist. In the case of an intentional 'relation,' however, the object-relatum need not exist.
3. For Brentano, then, all and only mental phenomena are intentional. Equivalently, all and only conscious phenomena are intentional. Brentano uses 'consciousness,' 'mental phenomenon' and 'mental act' as synonyms. (Psychology from an Empirical Standpoint, p. 102) Note that a mental act is not the same as a mental action, although there are mental actions. If I 'rack my brains' to remember someone's name, perhaps by running through the alphabet (Al, Andy, Bill, Bob . . .) then I am engaging in a mental action; but if I merely note the shape of an object, that is a mental act. Every mental action is a mental act, but not conversely. Act has the connotation of occurence, actuality, as opposed to something merely potential or dispositional.
4. Colin McGinn (The Problem of Consciousness, p. 33) writes that "Brentano's thesis was that all consciousness is intentional . . ." but goes not to mention a "converse Brentano thesis . . . that all intentionality is conscious." I think this shows a misunderstanding on McGinn's part. Brentano held both theses: intentionality is both necessary and sufficient for mentality. Indeed that is why took intentionality to be a mark or criterion of the mental.
By the way, it is quite common for analysts in the Anglosphere to invoke Brentano while betraying virtually no knowledge of what he actually thought. The attitude of Searle is widespread: "My method in philosophy is to try to forget about the history of a problem and the traditional ways of thinking about it and just try to state the facts as far as we know them." (Mind: A Brief Introduction, p. 111) I would only add that one cannot forget what one has never learned. I can't be sure, but my impression is that Professor Searle has not done much historical 'homework.'
5. Are there non-intentional mental states? If all consciousness is intentional, then what do we say about pleasures and pains? To be in pain is to be in a mental state, a conscious state. But there is no clear sense in which my present headache pain is about anything. It does not present an object distinct from itself in the way a memory of the Boston Common presents the Boston Common. Of course, it has a cause, dehydration perhaps, but it is not about dehydration. It is what it is and can described regardless of its cause or even whether it has a cause. By contrast, an intentional state cannot be described without describing its object.
On this question, a commenter wrote,
. . . a bodily sensation of pain has a body or a bodily state as its intentional object, whether the subject has a body or not. But maybe that's wrong: there is no intentional object of my thought that there is nothing.
I think that is wrong: headache pain is not of or about my head or my brain or any part of my brain, and could conceivably occur even if I had no body or brain. Even if the latter is not possible, it is conceivable, and this suffices to show that "intentional inexistence of an object" is not essential to a pain's being a pain in the way it is essential to an intentional state's being an intentional state. See the category Intentionality for more on this topic.
6. The thought that there is nothing. The commenter raises an interesting question here. Does this thought take an accusative or not? Brentano does not admit propositional accusatives given his (Aristotelian) hostility to Platonica, idealia, abstracta. But suppose we admit (Fregean) propositions as the direct objects of some intentional states. The thought that there is nothing can be understood as the entertaining of the proposition, Nothing exists. Entertaining, whether or not with hospitality, is undoubtedly an intentional state. Now Nothing exists though false, and indeed necessarily false, exists! That is, the Fregean proposition exists. (If it didn't exist it couldn't have a truth-value.) Therefore, the thought that there is nothing does, pace mycorrespondent, have an intentional object. It is just that the intentional object in this case is a false proposition.
Note also that the existence of the proposition Nothing exists is proof that Nothing exists is false. Indeed, given that the proposition exists in all metaphysically possible worlds, its existence shows that Nothing exists is necessarily false.
In sum, my commenter entertains the right view (that there are non-intentional mental states) for the wrong reason.