First used in 1914 in "Remembering, Repeating, and Working-Through," Freud introduced the term "acting out" to describe behaviors that have been repressed regarding others, but are manifested with the therapist. For instance, Freud's case study of a patient called "Dora" recorded how she abruptly ended her therapy sessions:
...because of the unknown quality in me which reminded Dora of Herr K., she took her revenge on me as she wanted to take her revenge on him, and deserted me as she believed herself to have been deceived and deserted by him. Thus she acted out an essential part of her recollection and phantasies instead of producing it in the treatment.
The curious thing about this "acting out" is that the patient is not aware of anything as part of memory or repression, but unconsciously repeats the action. It is a transference behavior, disruptive or inappropriate actions brought about by the pressure of unconscious desires.
On the other hand, "acting in" occurs outside the realm of therapy, existing in such areas as love relations, work relations, friendships. "Acting in" can be verbal responses, rather than physical. But, both "acting out" and "acting in" are responses to that which has been repressed.
Alan Rowan. "The Place of Acting Out in Psychoanalysis." Psychoanalytische Perspectieven, 2000.