I simplify this subject, of course, to a tool you can wield in many occasions. The learned socratists will explain otherwise the subtleties of the elenchus refutations. My pick from the Socratic myth is a minimalist, impressionistic account of a style and attitude I found most valuable to use for the everyday.
I use the Socratic wand when I want to highlight and identify what is actually discussed in a relevant occasion, to verify what people understand and mean, what they say about important subjects, and when I want to cause or help people discover that they only think they know but they know almost nothing. I found that this is the case in almost every domain, exactly as Socrates observed in his time. You would use it with caution, when adequate, not like an apprentice sorcerer.
To imitate Socrates, you need to have the self-insurance to announce with bold, stressed out humility, particularly when you are known to be knowledgeable, the teacher or even the expert: "Excuse me, I don't know much about this thing you speak about, can you please explain what it is, what it means so that we talk usefully?" Such questioning causes people to become aware of their ignorance, sometime even exposes them. You bring out into the open - do it if needed, tactfully if possible - that in fact we all know very little about the subject at hand, but we consider it obvious; after a few questions asking what and why (my favourite is to ask three times why), shared ignorance becomes obvious and puzzling; then, the door is unlocked for all to listen and to debate critically.
Without finding out that they speak most of the time without having acceptable definitions, people are complacent, sufficient, like cups full to the brim, incapable to absorb something different or new. You must help empty some of their their cup and make them feel that they "know nothing" [**]
Like Plato's legendary Socrates, you will make one learn - not by teaching lessons, not from received ideas and words - but being pulled from one puzzlement to the next, discovering more and more unexpected perspectives, feeling the uneasy borderline of what we do not know yet; learning advances not from tabula rasa to the right answer, not even from some certainty to more certainty but from question to question.
Deeper than erroneous definitions, one of the killers of good judgement and of productive conversation is the supposition that "we know" and that the words mean, naturally, the same thing for all of us; the best way to misunderstanding is to ask the inept question: "Did you understand?" and to take the empty answer: "Yes." for a proof of success. Meaning-full communication is on the contrary to ask: "What did you understand?" or "How would you define or explain this? What do you think about ...?" or some similar polite version (like "Please tell me more" or "Please explain") for finding out what was meant and understood.
For this reason, reject the lazy convention that everybody knows, “of course we all know the subject” and propose, with or without irony that you personally don’t know, that in fact you feel totally ignorant and want to be explained.
When you ask the Socratic Quid question  “What is it?”, persist politely; repeat the question if needed, to have clarified what it is and what it is not, what is different from other things. Do not accept the replacement of definitions by that which is essential and unique with enumerations of obvious examples.
Amazingly, most people, particularly authorities and experts, prove unable to give a true definition of things they profess or practice every day, as the deeper understanding is obscured with plenty of words, precision, details and received opinions, while essence is left for later or for the philosophers. It is hard to pin down the nature of things. We mainly know how they look and how we use them.
In fact the basic, seemingly self-evident looking notions we use every day, are the most difficult to define. Probably, to be defined, they require a point of view "out of the system" and higher order rules. But this - exact, correct definition - is not the real aim of the Socratic attitude as I value it; the aim is learning to think and learning to learn, learning to navigate the immensity of our ignorance without panic and without trying to reduce it to our received knowledge.
The second face of the Socratic myth as I retain it, is that method will not be found in in this or that question but in the questioning; it is not the trees, it is the Socratic forest.
Socrates the emblematic critical thinker gives himself the unrestricted and perilous licence to ask questions to the face of habit, power and pretence. He felt that to be free, to have some power of action, you must first know yourself and own the words you use; that unexamined life is not worth living . He took the common place exhortation from the front of the temple at Delphi "gnōthi seautón = "know thyself"" and turned it into a powerful rule. This knowledge only, makes us masters of ourselves, which is the precondition to rule other people. Without this effort of humbly knowing oneself no one can imitate Socrates, no philosophically correct questions will suffice..
Socrates (or maybe Plato) assumed that all the knowledge was already there , in the person, sleeping or gestating; the task of the master is, accordingly, the art of the midwife, to help people's own understanding to be born. At a most vulgar everyday level, in my teaching and my work with groups of people I found that indeed, if you ask and provide sufficient time and listen too, even the simplest audience will provide collectively some useful account of the issue at hand; sufficient sound material to facilitate the community to build up useful answers, deliberations, decisions and solutions which they accept and own, because they produced them. Certainly there is plenty of occasions for the skilled teacher or consultant or advisor, to contribute solid expertise and wisdom, with discretion but also decisively. Read the stone-soup story to see what I mean!
In fact, people know a lot
After that first step which was to make a learner or an audience understand that they did not understand (or too little) of things they believed to know well, this manner of Socrates, the maïeutic, the art of the midwife, demonstrates the contrary - that we know more that we think we know. Moreover, it makes us feel that we are not blank slates to be written on by other people, but instead we have the means to think with our own head. This is far of course from Plato's magnificent world of ideas waiting to be recalled; however the debased metaphor I draw is quite valuable for any teacher in our lowly real world.
With his manner of midwifery, after clarifying as much as possible "What it is? » Socrates leads the person, question after question, maybe less towards discovering truth than diminishing their mistaken beliefs: Maybe his "victims" learn less about the world, but certainly about themselves.What do I know? He often leaves them puzzled...in the company of ignorance. Learning to cope honestly with ignorance is itself precious learning.
I would add that after finding out what you believe and understand, your limits, where your mind is, (and only then) you can actually proceed to learn, and change.
The deepest face of the "Method of Socrates" as I see it, the gift I feel I received from him, is the handing over of the "method" itself, that Socratic way of being helping us in learning how to learn; Socrates teaches by example an attitude to knowledge, he teaches questions not answers, a way to examine everything, his magic pointing finger. An old Chinese story  illustrates best what I desire from Socrates:
An immortal hovering along a deserted road met a miserable half-starved beggar. In pity, the benevolent ghost touched a pebble with his finger and miraculously, the stone turned to gold. But the beggar did not look content.
Amazed, the spirit chose a bigger stone, laid his finger and Lo! it became gold too. The mendicant was still not satisfied.
Puzzled, the celestial being tapped a big rock into gold; the beggar grew visibly discontent. Exasperated, the immortal asked:
"What more do you want?"
The method of Socrates is what he does, the example he gives.
[*] The philosophers, expert in the study of Socratic thought, teach something else, much more abstract, which I am certainly not competent to debate. Theirs may be the true Platonic Socrates who actually was; I only care for one legendary tool from the ages-improved heritage as it trickles down to us, possibly myth, but a legend of value to my life and know-how.
[**] "I thought to myself, " I am wiser than this man ; for neither of us really knows anything fine and good, but this man thinks he knows something when he does not, whereas I, as I do not know anything, do not think I do either. I seem, then, in just this little thing to be wiser than this man at any rate, that what I do not know I do not think I know either."" Plato, Apology, in Plato I, Loeb CL, Harvard UP, Cambridge.., p.83 (Tr. Fowler)
 Plato, Meno (Tr. Guthrie) 71b, Bollingen ed., Princeton, 1989, p. 354: “..how can I know a property of something when I don’t even know what it is?”
 Plato, Apology, 38a
 Plato's Socrates seems to believe that knowledge comes from a divine realm of pure ideas and that it is pre-existent in man. I do not believe this, but there is always some pre-existent knowledge; I experienced many times the common-sense fact that when you meet real-life persons be it children or, even more, adults, they do contain rich previous experience and lore which must be considered and put to work, not denied or replaced. You cannot erase that reality and replace it with your teaching; you must build on it, from it.
 Fung Yu-lan, A Short History of Chinese Philosophy, Macmillan, NY, 1960, p. 330