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

It's the same for professional as well as amateur actors - I've known some pros that have a hard time reaching the crying point.

One technique includes allowing the actor to focus on the saddest memory they have, in solitude, then pull them out for the take. I recall seeing a documentary that showed Gary Oldman looking through a book of photos of his family (he had just split from Uma Thurman) on the set of Dracula that got him to the point of sorrow that Coppola wanted.

I have worked with some amazing amateur actors that would work themselves up into a state before filming. One actress imagined something terrible happening to her sister, and she delivered a very powerful performance (that got the crew all choked up).

If all else fails, the vapors from an onion can induce tears (but also cause redness), or a few drops of glycerin in the corners of the eyes can be released - but nothing beats real tears.

I have also heard of directors bullying/belittling actors to the point of melt-down, just to get the shot they want. But that's a pretty extreme way to go about it.

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

Not wanting to detract from Nobby's excellent answer I's like to add that very often directors don't use techniques in that sense, but just rely on their actors to know what they're doing.

Two days ago I saw an interview with Yorick van Wageningen about a rape scene with Rooney Mara in The Girl with the Dragon Tattoo. I was impressed by this young actress's professionalism. Contrary to what actors usually do, meticulously go through a scene prior to shooting it, she told him not to talk about it in advance, but to just let her suffer the rape as if it really happened to her. He said that between the cuts she was still crying.

Many actors can turn tears on and off at will. Others struggle with it and need the right kind of assistance and sometimes get it.

Jackie Cooper (1922-2011) in Please Don't Shoot My Dog (1982) tells how they pretended to shoot his dog to get him to cry for a scene in Skippy (1931).

David Hollander (born 1969) told how once when he was too happy to do a crying scene he made himself unhappy enough by destroying one of his treasured toys.

When you see a scene with crying or other intense emotions you have no way of telling how easy or hard it was for those particular actors to act that scene.

In an interview with Jimmy Kimmel, director Wes Craven (known for Nightmare on Elm Street and Scream) said he would tell a young actor "Your mother's dead" and then quickly walk off the set. Said it worked every time.

Some actors are just naturals, as other answers have mentioned. Here's Henry Thomas's audition tape for the role of Elliot in E.T. The tears start to flow right at the 2:00 mark.

As someone who does community theatre and has certain emotional triggers, I really channel into the things that set me off and work with them. The best one is songs, songs that are sad and that I know make me cry. I listen to them a few hours before going on stage and during my performance, replay the song in my head and relate it to how my character is feeling. This generally sets me off to the point where after I got off stage, I had other actors asking me how I did it and if I was alright.

What I am trying to say is, the key to acting is how you can trigger your emotions and what pushes you buttons. That is how you can get the best out of yourself and create a convincing performance.

I developed the ability to cry on command. I can keep a straight face and just let tears pour out so I guess it's easier for me.

One method that tends to work is yawning. If you can yawn while keeping your mouth shut your eyes will tear up. So being tired will help with those emotional scenes.