The influential Austrian philosopher Franz 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.)
What is intentionality? ‘Intentionality’ is Brentano's term of art (borrowed from the Medievals) for that property of mental states whereby they are (non-derivatively) of, or about, or directed to, an object. Such states are intrinsically such that they 'take an accusative.' The state of perceiving, for example is necessarily object-directed. One cannot just perceive; if one perceives, then one perceives something. The idea is not merely that when one perceives one perceives something or other; the idea is that when one perceives, one perceives some specific object, the very object of that very act. 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, or may or may not be true in the case of propositional objects. 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." It is also why Husserl can 'bracket' the existence of the object for phenomenological purposes. Intentionality is not a relation, strictly speaking, though it is relation-like. This is an important point that many contemporaries seem incapable of wrapping their heads around.
There are some interesting points of analogy between intentionality and potentiality. An intentional state exhibits
a. directedness to an object
b. an object that may or may nor exist
c. an object that may be, and typically is, indeterminate or incomplete.
For example, right now I am gazing out my study window at Superstition Mountain. The gazing is an intentional state: it is of or about something, a definite something. It takes an accusative, and does so necessarily. The accusative or intentional object in question presumably exists. But the intentional object is what it is whether or not it exists. The phenomenological description of object and act remains the same whether or not the object exists. Moreover, the object as presented in the act of gazing is incomplete: there are properties such that the intentional object neither has them nor their complements. Thus, to a quick glance, what is given in the intentional experience is 'a purplish mountain.' Just that. Now anything purple or purplish is colored, and anything colored is extended; but being colored and being extended are not properties of the intentional object. No doubt they are properties of the mountain itself in reality; but they are not properties of the precise intentional object of my gazing, which has all and only the properties it is seen to have. Furthermore, in reality, yonder mountain is either such that someone is climbing on it or not; but the intentional object of my momentary gazing is indeterminate with respect to the property of being climbed on by someone.
The potentiality inherent in a thing exhibits
a*. something analogous to intentional directedness: a potentiality is a potentiality for something, or to something. For example, a human embryo has the potentiality to develop, in the normal course of events, into a human neonate. But a human sperm cell lacks this potentiality. It has a different potentiality: it can combine with a human egg cell to form a zygote. A thing cannot just have a potentiality: every potentiality is a potentiality for something or to something. This something is not merely a something or other, but a definite something, analogously as in the case of intentional directedness.
b*. something analogous to the feature of an intentional experience whereby, from the occurrence of the intentional experience, one cannot infer the existence of its intentional object. Just as the intentional object may or may not exist without prejudice to its being an intentional object, a potentiality may or may not be realized. The embryo's potentiality to develop into a neonate may go unrealized -- and this without prejudice to the potentiality's being something quite definite and quite real.
c*. something analogous to the incompleteness of intentional objects. To revert to a hackneyed example, an acorn has the potentiality to become an oak tree. But this is not to say that there is some perfectly determinate (definite) oak tree that an acorn has the potentiality to become, a 50 foot oak tree the diameter of whose trunk is two feet, etc.
The same points can be made about dispositions. If a piece of glass is fragile, then it is disposed to shatter if suitably struck. There cannot be a disposition that is not a disposition to do something, to shatter, or explode, or melt. Second, the reality of a disposition is independent of its manifestation: a fragile piece of glass is fragile whether or not it ever breaks. From the fact that x is disposed to F one cannot infer that x ever Fs. This parallels a feature of intentionality: from the fact that x is thinking about Fs one cannot infer that there exist Fs that x is thinking about. (If I am thinking about unicorns it does not follow that there exist unicorns I am thinking about; if I want a sloop it doesn't follow that there is a sloop I want; if Ernest is hunting lions it doesn't follow that there are any lions he is hunting.)
Third, although a manifested disposition is a fully determinate state of affairs, this complete determinateness is not present in the disposition qua disposition. The disposition to shatter if suitably struck is not the disposition to shatter into ten pieces if suitably struck, although it is of course the disposition to shatter into some number or other of pieces, the exact number being left indeterminate.
Now here is a tough question: are dispositionality and intentionality merely analogous, or can we take it a step further and say that utimately there is no difference between dispositionality and intentionality? If that case could be made, then Brentano would be shown to be wrong in his claim that intentionality is the mark of the mental. For if the three characteristics of intentionality mentioned above are found below the level of mind in the physical world, then it looks as if intentionality cannot be the mark of the mental. Or should we stay instead that, since intentionality is the mark of the mental, and intentionality is found in nature below the level of mind, that there is something mind-like about all of nature?
These are just (half-formulated) questions.