The Scholastic theory that there are intentions is well known, both by those who accept it and dispute it; but the general points of the theory that the Scholastics used to explain them are much less well known.
Prior to the idea of intention is the idea of form. Our idea of a form starts off as an idea of a shape, and shapes are best known to us by the shapes that we make ourselves (since such forms take their truth from us), which is why the perennial example of form will always be the shape of a statue. Form determines something to be this or that – the statue starts existing when the sculptor stops forming.
The theory of intention arises in comparison to this: just as the subject of acting (like stone) can come to be this or that, the subject of knowing (a sense organ, an intellect) can come to be this or that. Since form is what determines the first activity, to word extends to mean what determines the second activity. Both actions are receiving form, but this reception is partly the same and partly different. The first difference between the two is that while the first form makes something be a new thing the second form allows both things to remain what they were before. The first form makes something be while the second doesn’t. This is why the Scholastics (until Suarez, at least) divided the second sort of form from the form that gives being or existence, so the intentional form was opposed to the entitative form (it was perhaps called “intentional” since it was not entitative but was directed to the entitative). The two forms related to different subjects too; and since the subject of the first form was called matter, the subject of the second was called immaterial.
The objections to this theory reduce to the claim that there is no difference between the first form and the second; that is, that knowledge is simply an entitative change. Suarez made this argument popular, and so gave the first version of the modern physicalist argument against intentions (though Suarez was emphatically not a physicalist). The objection is simple: seeing green is simply a more complex entitative change.
The argument about intentions was, for the Scholastics at least, a dispute about whether there was any real difference between knowers and non-knowers. Our modern dispute comes to this too, though contemporary people lack the notion of form that makes this clear.