In philosophy, "Intentionality" is usually understood to mean aboutness or directedness. A mental state is an intentional mental state (has intentionality) when it is about, or directed at some object. Examples:
Beliefs are necessarily intentional mental states. It is impossible to have a belief without the belief being about something. This is the subject matter or the content of the belief. So even if you believe that nothing exists, there is a sense in which the belief is still about something, in that it has a certain content.
Apart from beliefs, our wishes and desires, hopes and fears, are also intentional mental states. I might hope that it will rain tomorrow, and maybe you fear that it will rain tomorrow. My hope and your fear are different mental states, but they have the same content, that it will rain tomorrow.
Perception is also intentional. If you are looking at a bird, your perceptual experience is directed at the bird. The same goes for mental imagery as when you imagine a mountain made of ice cream. There is no such mountain in the real world, but your imagination has a content nonetheless. This example tells us that an intentional mental state can be about something that does not actually exist. A little girl might believe that Santa Claus lives in Finland. Her belief is about Santa Claus even though Santa Claus does not exist.
Pains and itches are conscious mental states. Philosophers these days like to say that a conscious mental state is a mental state for which there is something it is like to be in it. If a creature is having a conscious mental state, there is something it is like to be that creature. Whereas there is nothing it is like to be a stone. There is, however, something it is like to be in pain, and what it is like to be in pain is what is distinctive about pain. There is also something it is like to have an itch, and it is different from what it is like to be in pain. Conscious mental states are also called phenomenal states. They are said to possess qualia. One of the most puzzling problems in philosophy is to explain how a physical object can possess qualia, or consciousness.
Mental imagery is one example. If you imagine yourself flying above the ocean presumably you have some kind of conscious experience. It is a bit like having a picture in one's mind. On the other hand this state of our imagination has content : it is about you flying above the ocean. Or consider a different example. Suppose you feel very angry at your boyfriend. This is again a conscious mental state since there is something it is like to feel anger. But your anger is also an intentional state since it is directed at your boyfriend. Your state of anger is a state about him.
It seems clear that there are lots of intentional mental states that are not conscious. We have lots of different memories, and most of the time we are not actively recalling them. For example, you remember your name and this memory is an intentional mental state. But you were probably not thinking about your name five minutes ago. Yet at that time it is true that you remember your name, even if you were not thinking about it and was not conscious of your memory. So here is an intentional mental state that is not a conscious mental state. Similarly, you do not cease to believe that 1+1=2 when you are sleeping, but there is nothing it is like at that time to have the belief.
Psychology and cognitive science have also discovered lots of mental processes in our brain. Some of these processes help us recognize faces, others help us identify the direction of sounds. They involve information processing so it is plausible to think that these are mental processes that include intentional mental states. But we are usually not aware of these mental processes happening. We just know that a sound is coming from the left but we are not aware of how we come to know this. So lots of these low-level but complicated mental processes are also unconscious but intentional.
And if Freud is right, then perhaps even some emotions can be unconscious. Perhaps it is possible for unconscious anger toward a person to manifest itself in nasty behavior toward that person, but the one who is angry actually does not feel the anger.
This is actually quite a controversial question. Some philosophers think that there are non-intentional mental states. They argue that sensations such as pains and itches are not about anything at all, and so they are not intentional.
It certainly seems strange to speak of a pain as being "about" something. On the other hand, it does not seem strange to say that pains and itches are states that carry information. We might say that a pain carries information about what is happening to our body, for example. In cases of phantom limbs, patients who have their limbs amputated might continue to feel pain which they locate in body parts that no longer exist. One way to understand this is to say that their pains are misrepresentations in that they carry false information about their bodies. This is somewhat similar to having false beliefs. So if we understand intentionality as carrying information then perhaps pains and itches can be seen as intentional too.
However, if this is the definition of intentionality, we might not be able to say that desires are intentional states. We do not normally say that desires carry information. If I desire sugar my desire is about sugar, but the desire does not carry any information about sugar. Perhaps we can define an intentional mental state as any mental state that either is about something or carries information. More generally, we might say that intentionality is a matter of having content. Then beliefs, desires and pains can all be regarded as intentional mental states.
This issue relates to the question of whether there is "a mark of the mental". This is just the question of whether there is any distinctive feature that is unique to all mental states, a feature that all and only mental states possess. Perhaps intentionality is the mark of the mental. On the other hand, perhaps the concept of the mental is vague and there is no precise definition of mentality.
Sometimes a distinction is made between original and derived intentionality. Sentences, pictures and diagrams are also intentional objects since there are things they might be about. But their intentionality derives from our minds and so they have derived intentionality, whereas our mental states have original intentionality.