in philosophy, the difference between "small r" real/reality and The Real is a distinction made necessary by the limitations of language that muddy the separation between the Imaginary (a personal/subjective/changeable "reality" that exists for the individual/group that is pondering/humoring it) and The Real (the category of True Statements about the world, unfettered by sense perception or symbolic/cultural interpretation).

for example: the phrase "the sky is blue" is a statement made about a subjective experience of reality; for it to be a True statement, the subject who says it must either a) be looking at a sky that is blue and validating this blueness by their own sensory perception of the wavelengths of light they see, or b) have previous subjective experience of blue skies, which they attempt then to relate to another. Note, of course, that the blueness of the sky is not supported by any concrete, objective reality beyond the fact that the physical world can align in such a way that sometimes the sky is percieved to be blue.

On the other hand, statements made within the space of The Real would be True in an objective way, regardless of the personal experiences of the subject making the statement. From the wiki, explaining the Lacanian interpretation:

The order of the Real is not only opposed to the imaginary but is also located beyond the symbolic. Unlike the symbolic, which is constituted in terms of oppositions such as "presence" and "absence", there is no absence in the Real. The symbolic opposition between "presence" and "absence" implies the possibility that something may be missing from the symbolic, the Real is "always in its place: it carries it glued to its heel, ignorant of what might exile it from there." If the symbolic is a set of differentiated signifiers, the Real is in itself undifferentiated: "it is without fissure." The symbolic introduces "a cut in the Real," in the process of signification: "it is the world of words that creates the world of things." Thus the Real emerges as that which is outside language: "it is that which resists symbolization absolutely." The Real is impossible because it is impossible to imagine, impossible to integrate into the symbolic order. This character of impossibility and resistance to symbolization lends the real its traumatic quality.

ok but what does THAT mean? The Real is traumatic in its impossibility? Well, lets go back and try to make a Real statement about the sky. The sky is blue doesn't work, because that subjective experience implies the "presence of blueness" that precludes the "absence of not-blueness." It is inadequate as a Real statement because the sky can be one of any number of colors, patterns, conditions, etc. In this same way, what the weather is, or the constituent matter making up the gas admixture of our atmosphere, these questions can only ever be answered through subjective processes, not just because any statement made about these attributes of the sky will be subjective, but also because the words themselves are from the world of words, created by subjective human social experience, to explain the world of things experienced by that society. The degree to which statements about these attributes could be Real is hindered by the absolute reality that the statement itself is trapped in the subjective realm from its very conception! A statement about the sky that is Real has to be configured to make sense outside of this subjective world of words and things described by them. One that gets close, but is still hindered (ever so slightly) by the fact that we're still calling it "sky", would be something like, "The sky IS." i.e. whatever subjective interpretations of reality are caught up in the meaning of the word "sky", there is an objective Truth that that thing described by the word "sky" is definitely up there, is definitely present in Reality.

But wait, that's boring. Where's the "traumatic Real"? To that, I will first remind you of Chicken Little, and now I will subject you to a dramatization of the consequences of The Real:

ahem

JESUS FUCKING CHRIIIIIIIST WHAT IS THAT THING!?!?!? AAAAAAAAAAAAAAAAAUGH FUCK ITS EVERYWHERE AND ITS JUST HANGING UP THERE! THEY TOLD ME IT WAS JUST A BLUE THING THAT'S BIG AND SOMETIMES ITS WET BUT AN ICE CUBE THE SIZE OF MY FIST JUST CAVED IN THAT DUDES SKULL AAAAAAAAAGHGHGHGHGH WHAT THE FUCK IS THAT LOUD NOISE WHAT ARE THOSE FLASHING LIGHTS WHAT AM I SUPPOSED TO DO WHERE AM I SUPPOSED TO GO HOW WILL MY PEOPLE SURVIVE THIS TERRIBLE CONFUSING CONSTANTLY PRESENT FICKLE MONSTER OF REALITY THAT CAN KILL INDISCRIMINATELY AND TAKE WHOLE HARVESTS OVERNIGHT PLEASE LORD GOD SWEET BABY JESUS OR MAYBE THE KING OF THE LAND DO SOMETHING AUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUIEEEEEEEEEEEEEEEEE!

Okay okay, definitely a dramatization, but hopefully it gets the point across: the Real has consequences that are generally invisible to/hidden by the world of subjective language/interpretation, consequences that cannot simply be explained away by developing more sophisticated labeling conventions; the system of labeling will never be sufficient to paper over the raw nature of the underlying physical reality.

Anyway, this is all loose, and probably a bit wrong in places, but it gets the basics of the concept across.

Now go read Capitalist Realism again :)