An alief is a belief-like attitude, behavior, or expectation that can coexist with a contradictory belief. For example, the fear felt when a monster jumps out of the darkness in a scary movie is based on the alief that the monster is about to attack you, even though you believe that it cannot.
The word derives from a sort of pun on dual process theory; what beliefs ("B-liefs") are to system 2, aliefs ("A-liefs") are to system 1. Thus, beliefs are explicitly held beliefs which inform slow reasoning, while aliefs are implicit attitudes which guide fast reactions. However, dual process theory is not totally necessary to make sense of the term alief.
A related pun is "cesire vs desire"; a desire ("D-zire") is an explicit want which enters into explicit planning, while a cesire ("C-zire") is an implicit one which guides reactions....