The word noema (plural: noemata) derives from the Greek word νόημα meaning "mental object". The philosopher Edmund Husserl used noema as a technical term in phenomenology to stand for the object or content of a thought, judgement, or perception, but its precise meaning in his work has remained a matter of controversy.
Husserl's noema
In Ideas: General Introduction to Pure Phenomenology (1913), Husserl continued and built on the (ancient to modern Greek to early modern German Idealism philosophies') terms "noema" and "noesis" to designate correlated elements of the structure of any intentional act—for example, an act of perceiving, or judging, or remembering:
"Corresponding to all points to the manifold data of the real (reelle) noetic content, there is a variety of data displayable in really pure (wirklicher reiner) intuition, and in a correlative 'noematic content,' or briefly 'noema'—terms which we shall henceforth be continually using."
Every intentional act has noetic content (or a noesis—from the Greek nous, "mind"). This noetic content, to which the noema corresponds, is that mental act–process (e.g., an act of liking, of judging, of meaning, etc.) which becomes directed towards the intentionally held object (e.g., the liked as liked, judged as judged, or meant as meant). That is to say, every act has, as part of its formation, a noematic correlate, which is the object of the act—that which is intended by it. In other words, every intentional act has an "I-pole (the origin of the noesis)" and an "object-pole (or noema)." Husserl also refers to the noema as the Sinn or sense (meaning) of the act, and sometimes appears to use the terms interchangeably. Nevertheless, the Sinn does not represent what Husserl calls the "full noema": Sinn belongs to the noema, but the full noema is the object of the act as meant in the act, the perceived object as perceived, the judged object as judged, and so on. In other words, the noema seems to be whatever is intended by acts of perception or judgement in general, whether it be "a material object, a picture, a word, a mathematical entity, another person" precisely as being perceived, judged or otherwise thought about.
Interpreting Husserl
In fact, commentators have been unable to achieve consensus on exactly what a noema is. In a recent survey, David Woodruff Smith distinguished four different schools of thought. On one view, to say that the noema is the intentional object of an act of consciousness is to mean that it quite literally is an object. Husserl's student Roman Ingarden, for example, held that both ordinary objects, like chairs and trees, and intentional objects, like a chair precisely as it appears to me, or even a fictional tree, actually exist, but have different "modes" of existence.
An alternative view, developed primarily by Aron Gurwitsch, emphasizes the noema of perceptual experience. Most ordinary objects can be perceived in different ways and from different perspectives (consider looking at a tree from several different positions). For Gurwitsch, what is perceived in each such act is a noema, and the object itself—the tree, say—is to be understood as the collection or system of noemata associated with it. This view has similarities with phenomenalism.
Robert Sokolowski, alternatively, holds that a noema is just the actual object of perception or judgment itself, considered phenomenologically. In other words, the noema of the judgment that "this chair is uncomfortable" is neither an entity (the chair considered as uncomfortable) which exists in addition to the chair itself (but with a different mode of existence)—the Ingarden view; nor is the noema of such a judgment identified with a particular tactile perception of the chair—which along with other perceptions constitutes the chair as such—the Gurwitsch view. For Sokolowski, the noema is not a separate entity at all, but the chair itself as in this instance perceived or judged. This seems consistent with Husserl's emphasis on the noema as the "perceived as such…remembered as such...judged as such..."