Socrates was one of the greatest educators who taught by asking questions and thus drawing out answers from his pupils ('ex duco', means to 'lead out', which is the root of 'education'). Sadly, he martyred himself by drinking hemlock rather than compromise his principles. Bold, but not a good survival strategy. But then he lived very frugally and was known for his eccentricity. One of his pupils was Plato, who wrote up much what we know of him.
Here are the six types of questions that Socrates asked his pupils. Probably often to their initial annoyance but more often to their ultimate delight. He was a man of remarkable integrity and his story makes for marvelous reading.
The overall purpose of Socratic questioning, is to challenge accuracy and completeness of thinking in a way that acts to move people towards their ultimate goal.
Get them to think more about what exactly they are asking or thinking about. Prove the concepts behind their argument. Use basic 'tell me more' questions that get them to go deeper.
Probing their assumptions makes them think about the presuppositions and unquestioned beliefs on which they are founding their argument. This is shaking the bedrock and should get them really going!
When they give a rationale for their arguments, dig into that reasoning rather than assuming it is a given. People often use un-thought-through or weakly-understood supports for their arguments.
Most arguments are given from a particular position. So attack the position. Show that there are other, equally valid, viewpoints.
The argument that they give may have logical implications that can be forecast. Do these make sense? Are they desirable?
And you can also get reflexive about the whole thing, turning the question in on itself. Use their attack against themselves. Bounce the ball back into their court, etc.