We naturally tend to take our experience for granted. One of the profound innovations of Kant’s Critique of Pure Reason was an emphasis on processes of synthesis at several levels in the formation of experience. Dominant medieval and early modern views had tended to assume some kind of direct, unproblematic mental uptake of experienced objects or data from appearances. Kant’s talk about processes of synthesis is the positive account that complements his famous rejection of this “dogmatism”. For him, all conceptual uptake involves judgment, and all judgment involves synthesis, or putting together many things.
Aristotle had hinted at something like processes of synthesis in his mention of a common sense responsible for correlating perceptions from different senses like sight and touch into what we might call sensory objects. This was slightly expanded upon by Alexander of Aphrodisias in the late 2nd or early 3rd century CE, but it was not until Kant that the idea of synthesis began to be developed more fully. Plato and Aristotle treated judgment and what we might call synthesis as a good deal more difficult and provisional than other authors in the intervening period.
The most famous kind of Kantian synthesis happens when concepts are applied to parts or aspects of the manifold of sensory and other intuition. Even in very simple cases, this turns out to involve many judgments, which in turn involve others. Kant associates this with reason and conscious activity.
Another kind of Kantian synthesis applies at a more elemental level to pieces or aspects of content in the manifold of intuition. This seems to be essentially unconscious. As Beatrice Longuenesse pointed out in her outstanding Kant and the Capacity to Judge (1998), Kant argued that this unconscious synthesis in intuition and the more famous one involved with concepts in thought both depend on the same top-level table of categories. One possible way to interpret this is that Kantian intuition incorporates and in its own way autonomously and unconsciously applies conclusions of previous judgments or processes of synthesis, based on some kind of primitive sense of similarity to current circumstances. Of course, this is not guaranteed to be sound, but the idea is that it works well enough in many practical situations, and can be also refined and corrected by conscious judgment. (See also Passive Synthesis, Active Sense.)
A third kind of Kantian synthesis is the synthesis of unities of apperception. This is a high-level combination of very many judgments or commitments in a way that respects coherence. With either the addition of a track record or the interpretation of commitment as the commitment reflected in actions, this is what constitutes the moral identity of a person, which does not come ready-made (but see Obstacles to Synthesis).
The theme of synthesis was extended by Hegel to describe purely logical processes, as well as what Brandom would call genealogical ones.