Wikipedia page is excellent
I first studied Madhyamaka at Naropa University in the spring of 1992. I remember it specifically because it provoked one of my first direct experiences of emptiness. I was talking on the phone with a friend in the class, a monk named Sherap, and the world dissolved completely. The sun was coming in the window of the dining room. I was sitting on the floor, holding the phone, talking through these Madhyamaka exercises, and the solidity, the separateness, of each of the objects in my experience disappeared. It was a beautiful day.
So I know that this classic Buddhist approach to cutting concepts can be extremely effective. See also: Meditation on Wisdom for a related exploration of emptiness and transcendent knowledge. This page is a big rough sketch of Madhyamaka logic. My hope is that one day I can communicate this view to modern American meditators in a clear and coherent way. So that they too can experience emptiness directly.
According to the classical Indian Mādhyamika thinkers, all phenomena (dharmas) are empty (śūnya) of "nature",[8] of any "substance" or "essence" (svabhāva) which could give them "solid and independent existence", because they are dependently co-arisen [9]
Nagarjuna's critique of the notion of own-nature[note 5] (Mk. ch. 15) argues that anything which arises according to conditions, as all phenomena do, can have no inherent nature, for what is depends on what conditions it. Moreover, if there is nothing with own-nature, there can be nothing with 'other-nature' (para-bhava), i.e. something which is dependent for its existence and nature on something else which has own-nature. Furthermore, if there is neither own-nature nor other-nature, there cannot be anything with a true, substantial existent nature (bhava). If there is no true existent, then there can be no non-existent (abhava).[30]
The central idea is that dharmas are empty of svabhāva.[15] This term has been translated variously as essence, intrinsic nature, inherent existence, own being and substance.[16][17][15] Furthermore, according to Richard P. Hayes, svabhāva can be interpreted as either "identity" or as "causal independence".
This idea of svabhāva that Madhyamaka denies is then not just a conceptual philosophical theory, but it is a cognitive distortion that beings automatically impose on the world, such as when we regard the five aggregates as constituting a single self.
But this "emptiness" itself is also "empty": it does not have an existence on its own, nor does it refer to a transcendental reality beyond or above phenomenal reality.
Tetralemma
It is substance-svabhāva, the objective and independent existence of any object or concept, which Madhyamaka arguments mostly focus on refuting.[19] A common structure which Madhyamaka uses to negate svabhāva is the catuṣkoṭi ("four corners" or tetralemma), which roughly consists of four alternatives: a proposition is true; a proposition is false; a proposition is both true and false; a proposition is neither true nor false.
As we take stuff apart - as we look for a kernel of identity, or the core thing that makes something what it is, we realize that any single thing is made up of smaller components down to infinity.
Take a steak, cooked medium rare. The whole steak, off the grill after resting for 10 minutes, is a steak. It is. Now take a sharp knife and cut into it. Juice comes out, you can see the inside of the meat. Is it still a steak without all that juice? Yes it is.
Cut it in half. Now you have two steaks, correct?
Cut off a slice of meat from one. Is that steak? Yes, it’s a smaller piece of steak, small enough to put in your mouth.
Now trim the bone from the steak. Is it still steak without the bone? Yes. The bone and the juice were part of the steak before, but now, without them, we still have steak. They were not the essence of the steak, even though they were part of the steak.
Now trim the fat from your slice of steak. The fat is not the essence of the steak. But it was part of the steak before. It’s clear that what we called steak actually had lots of component parts that when isolated are not steak. So our labels, our definitions of steak are actually pretty loose. They are not exact.