Intentionality, most simply, is the way in which our thoughts somehow “reach out” to objects in the external world when we think about them. An example of intentionality in action is as follows: you thinking about your neighbor’s dog. In this example your thoughts have somehow reached out to the neighbor’s dog, or at least some part of that dog seems to have found its way into your thoughts. Intentionality is a puzzling subject not because we are unsure if it exists, as the fact that you can think about your neighbor’s dog clearly demonstrates its existence, but because it is hard to see exactly how it could work. For example if we were dualists we might assume that the mental substance somehow extended itself to reach the neighbor’s dog. Alternatively theories have been postulated that special relations are part of the world, relations that connect your thoughts to the neighbor’s dog. Your mind is then somehow connected to these relations, allowing your thoughts to encompass the neighbor’s dog without leaving your body. Admittedly this is a poor characterization of such theories, but what it illustrates is that such theories are externalist, meaning that they postulate some kind of direct connection between your mind and something outside your mind, either the dog itself or the relation.
Materialists tend to reject these theories, and hold that the mind is not connected to the rest of the world except through the physical inputs and outputs of the brain. This assumption often leads to the following flawed theory of intentionality: that our thoughts contain representations of the objects they are directed out. Such a theory works for the most part, but there are cases where it disagrees with our experience. First consider that not only can our thoughts be about an object in the world, they can also be about another thought. In fact our thoughts can even be directed at themselves, for example the thought “This very thought is self-referential”. The problem with a representation theory is that such a thought would have to contain a representation of itself, which in turn would contain a representation of itself, ect. However because our minds are finite such a thought could not possibly exist, which conflicts with out experience of actually having the thought.
The solution to such dilemmas is to hold that thoughts contain “pointers” or “references” to the representations that have formed in our minds, and not the representations themselves. For example my perceptions of the neighbor’s dog have resulted in my forming a representation of their dog in my mind. When I have a thought about their dog my thought does not contain this representation but merely contains a reference to it. If I later have another thought about that dog that thought has a reference to this very same representation. Of course we don’t actually perceive the references directly; the actual mechanics are governed by the biology of our brain, meaning that we may feel phenomenally that the representation is part of the thought while in reality the biological implementation of the thought has separated it from the representation. This solves the self referential thought problem neatly, for now thoughts about thoughts only contain references to the other thoughts. A thought about itself could easily contain a reference to itself, allowing us to both have such a thought and not require an infinite mind.
I must of course mention now that this theory is based on ideas that are not solely my creation. Specifically the philosopher Brentano first developed the idea that intentionality might only reach out to other mental entities (first as far as I know). Secondly the idea that our thoughts might point to other parts of our mind instead of representing them was first put forward by David W. Smith in his essay “Consciousness with Reflexive Content” (again, as far as I know).