"Achrayut" is usually translated as "responsibility," though its connotations are subtly different. It is widely used across the Jewish world.
In the Orthodox communities, "achrayus" is employed colloquially as the religious responsibility you have to improve or rectify a given situation.
In modern Hebrew, "achrayut" is a much-used word in political and military circles, denoting the responsibility of commanders for the successes and costs of their decisions.
However, the respective roots of "responsibility" and "achrayut" illustrate their different connotations. "Responsibility," as the word suggests, is the extent to which you have to respond, answer or account for your decisions and actions.
"Achrayut" comes from the Hebrew word "acher," meaning "other." It refers to your moral commitment to the other in a given situation - not just to answer to the other for your actions, but also to make the other's needs and concerns your own.
The first use of "achrayut" occurs in legal Mishnaic texts. A guardian who has "achrayut" over property is responsible to replace the goods with others, if they are lost or stolen (Bava Metzia, 3:11.)
Land over which "achrayut" applies must be mortgaged to meet other obligations (Kiddushin, 1:5).
More recently, the 20th- century French Jewish philosopher Emmanuel Levinas (who died in 1995) made the notion of our "achrayut" to the other the cornerstone of his moral and religious thought.
The other always lies beyond our comprehension; he or she is irreducibly strange to us and cannot be absorbed within an intellectual system which claims to give us exhaustive knowledge of the other.
It is this very otherness which, for Levinas, places on us an immense responsibility to meet the moral demands which the other places upon us.
Being acher, ultimately unique and different from us, gives rise to our "achrayut" to meet the other's needs.