What is Lacanian Real

Some simplified answers copied from a comment thread.

 The order of the Real is similar to the remaining two orders (Imaginary and Symbolic) in the sense that all psychoanalytic phenomena may be described accordingly through or with these orders.
It has already been pointed out that the Real cannot be symbolized or be articulated into language, therefore we cannot speak of it. Lacan also describes the Real as "the domain of whatever subsists outside symbolization" (Ecrits, p. 388).
It would also be wrong to equate the Real with reality, since in Lacanian theory, reality is a combination of the Imaginary and the Symbolic. But it would be right to link it with Impossibility - impossible to imagine, to integrate into language, impossible to obtain. Therefore the Real has an essentially traumatic quality to it, precisely because of this impossibility and this resistance against symbolisation, that it is everything excluded from our reality.
In the case of Little Hans, Lacan writes, there are two Real elements (with its traumatic quality), disrupting the imaginary harmony: the real penis and his baby sister. As for objects, an object can lose its meaning to us and what is left is something mysterious and uncanny.
Don't know if it helped (people are very welcome to add or whatever), but I would recommend "Lacan, Nous et le Réel" (on youtube) by Christian Dubuis Santini. I am currently watching that as well and I think its really great.

 You can think of a flag. The symbolic is the country represented by the flag. The imaginary is all the qualities you attribute to the country,the Real is what is left when you take out the symbolic and the imaginary : a silly peace of rag, probably full of mites and invisible microbes.

 The rules of chess (the moves each piece can make, the board, etc.) are The Symbolic. The level on which each piece is shaped and given a name (King, Queen, pawn) is The Imaginary. And the contingent circumstances that are left out of these registers (the intrusion on the game by my dog that upsets the chess board) is The Real.