What do you mean by in the mirror world? I was trying an attempt to replicate Wu experiment mentally when I heard the term mirror world kept popping ups, why should we care what happens inside the mirror world? Is it a math thing?
 A: Yes, it's a math thing. However, since physics is about describing the world as it is, and since physicists use mathematics to "understand" the laws of nature, the implications matter: If we include this "thing" in our mathematics to describe nature, we alter our predictions about nature.
Symmetries and conservation laws are fundamental in physics, because they tell us much about nature. E.g. we "know" that the momentum is conserved in a collision of two particles, and we "know" that fermions have an anti-symmetric wave function. These laws help us to predictable the outcome of experiments and to cross-check our calculations: If I do some calculation and obtain a non-symmetric (neither symmetric nor anti-symmetric) wave function of a fermion, I better redo my calculation. The same is true if I try to formulate an equation: If we use an equation which is not invariant under Lorentz transformation, we "know" that it probably only an approximation -- I use quotations marks for the word know, because this is still only an educated guess. 
The same is true for parity. Parity is a fundamental symmetry. It is quite "natural" for us to assume that it is conserved. Thus, it was rather shocking when Wu published her finding and proved that the weak interaction does not conserve parity. 
In order to put this into some perspective, let's consider modern assumptions. Today, "we" believe that CPT (charge, parity, time) is a conserved quantity under inversion. It is a valuable assumption, because it helps to sort our list of theories according to their probability of correctness: A theory which does not conserve CPT is generally considered to be less probable true than a theory which conserves CPT. Thus, this CPT "math thing" has implications: Physicists around the world use it to modify their theories. 
The math of parity is defined here, while the Wikipedia article about Wu's experiment provides a simple example to the topic. Consider a 2D clock (left picture) and it's mirror (right picture)

In a world which conserves parity and were a "left clock" rotates clockwise, the mirror image of the clock rotates anti-clockwise, see upper image. However, if we would observe that the mirror image of the clock rotates clockwise as well, 

we would conclude that parity is not conserved. 
A: The mirror universe is a universe in which the arrow time is pointing opposite to the time arrow in our universe (in reality the mirror should be 4-dimensional). Both arrows start at the big bang. Things happening in that mirror universe are happening at the same time as things over here, but "on the other side" of the big bang. I'm not so sure that things happening in that universe are identical to things happening here. I don't hope so, because I don't believe there is an anti-copy of me somewhere. Though an anti-me could be fun!
Here is a picture of the evolution of our universe since the big bang.

When you put a mirror at the flashing bang (perpendicular to it), then you will see in this mirror, well, a mirrored universe. A mirror universe. Identical to ours but the arrow of time, space (together spacetime), and matter and anti-matter are in(re)versed. I think this doesn't correspond to reality though. We will never be sure though.
As an aside
It's my guess there does exist a kind of mirror universe, but with the mirror placed parallel to our universe. The arrow of time is pointed there in the same direction as in our universe.  
In the rishon model matter and anti-matter are present in equal amounts in our universe. In this mirror universe also, but nevertheless what we call here matter (quarks, electrons, protons, etc.), is "over there" anti-matter (anti-quarks, anti-electrons, anti-protons, etc.). Not because of the mirror but because of the rishons are anti-rishons over there. Things over there don't have to happen identically to things happening here, but the rishon content is the same.
And here, too, we never will be sure. We *can** however, investigate if rishons exist. 
A: Parity (aka reflection aka the mirror world) is when you switch coordinates from $x, y, z$ to $-x, -y, -z$.
This looks so much like a simple coordinate transform that we would instinctively expect it to lead to the same physics, except it doesn't.
