This is the ability to have internal states that refer to or are about entities or situations more or less remote in space or time, or even non-existent or wholly abstract things.
So intentional states include contemplating clouds, dreaming you are a duke, exploring equations, pondering a possible action, seeing a snake, or wanting to win someone's favours. These are all cases of awareness or consciousness of something, including hypothetical or impossible objects or situations. A sophisticated mind may also have thoughts or desires about its own state -- various forms of self-consciousness are also cases of intentionality.
All intentional states seem to require the existence of some kind of representation of the content of the state: some representation of whatever is believed, perceived, desired, imagined, etc. A major theme in AI is therefore investigation of different kinds of representations and their implementation and uses. This is a very tricky topic, since there are many different kinds of representational forms: sentences, logical symbols, computer databases, maps, diagrams, arrays, images, etc. It is very likely that there are still important forms of representation waiting to be discovered.