What techniques are used by directors to make an actor cry?

Wikipedia leads back to Constantin Stanislavski and the Stanislavski system, a system of acting techniques to access emotions. He prefered to refer to it as system (little 'S'). The system is the result of Stanislavski's many years of efforts to determine how someone can control in performance the most intangible and uncontrollable aspects of human behavior, such as emotions and art inspiration. The most influential acting teachers, including Richard Boleslavsky, Vsevolod Meyerhold, Michael Chekhov, Lee Strasberg, Stella Adler, Harold Clurman, Robert Lewis, Sanford Meisner, Uta Hagen, Ion Cojar and Ivana Chubbuck all traced their pedigrees to Stanislavski, his theories and/or his disciples.

Lord Laurence Olivier wrote that Stanislavski's My Life in Art was a source of great enlightenment" when he was a young actor.

Sir John Gielgud said, "This director found time to explain a thousand things that have always troubled actors and fascinated students." Gielgud is also quoted as saying, "Stanislavski's now famous book is a contribution to the Theatre and its students all over the world."

Stanislavski once insisted that all actions that a person must enact, such as walking, talking, even sitting on stage, must be broken down and re-learned. For example, his book, translated into English as "Building a Character," gives a description of the correct way of walking on stage. Such rigors of re-learning were probably constant throughout his life.

Stanislavski's system is a method for actors to produce realistic characters on stage. His original studies of techniques led to the use of 'Emotional Memory' that required actors to trigger the emotions of their characters internally.

The example of crying is specifically covered in the article. Stanislavski developed the "method of physical actions," to solve the dilemma of spontaneous emotion in a created environment. In this technique, the actor would perform a physical motion or a series of physical activities to create the desired emotional response for the character. Emotions were considered to be formed from the subconscious, so this technique allowed the actors to consciously target and control their subconscious emotions through movement. For instance, if an actor needed to weep, he could sigh and hold his head in his hands, a physical action that many who are crying instinctively do...

The correct physical action does not come automatically for every psychological response nor are they stimulate identical responses for every individual. Many times, actors need to experiment until they determine what best works for them and for the character they are trying to portray. The best way to experiment with this is through improvisation. The best improvisers are those who can intuitively act and behave onstage as though they are in a real situation.

Additional techniques are explained in the wikipedia article - an interesting read.
For the TLDNR crowd, here's a different take, a quick-and-dirty:
An Actor's Guide to Crying and Tears