Q:

What is the difference between the standard representants of $\mathbb Z/q\mathbb Z$? 

The symbol $\mathbb Z/q\mathbb Z$ (given that $q$ is prime) represents the prime field $\mathbb Z_q$. Basically, the elements of this field are represented by $\{0, 1, \dots, q-1\}$, let's call this the "normal representation".  However, in lattice based crypto, the elements of the field are represented by $\{-(q-1)/2, \dots, (q-1)/2\}$, let's call this the "lattice representation". 
What is the difference between these two representations?  Why is the second one used in lattice crypto?  Would it work if the normal representation was used instead? 
I would also be very grateful if someone could show an example of how $\mathbb Z_7$'s elements $\{0, 1, 2, \dots, 6\}$ are mapped into the other representation. 

A:

When decrypting in lattice-based cryptosystems, one computes a value $v \in \mathbb{Z}_q$ that is guaranteed to be congruent to a "small" integer $e \in \mathbb{Z}$, where $e$ encodes the message (e.g., as the parity of $e$ modulo 2).   By using the integer representatives between $-q/2$ and $q/2$, one can recover the small integer $e$ (and thereby recover the message) simply by "lifting" $v$ to its integer representative. 
If we instead used the representatives $0$ through $q-1$, then the representative of $v$ would often be a little smaller than $q$ (specifically, when $e$ is negative), and would not have the proper parity. 
Geometrically, using the correct representatives corresponds to "decoding" the lattice $q\mathbb{Z}$ under small error: given the coset $(v + q\mathbb{Z}) \in \mathbb{Z}_q$, we find the smallest element in the coset, which is $e$.
Many papers conflate $\mathbb{Z}_q$ with its set of representatives, but this can cause confusion about what operations are legal and meaningful.  Values that are known to be "small" should be seen as elements of $\mathbb{Z}$, which are then reduced mod $q$ when combined with values from $\mathbb{Z}_q$. When we later want to recover a small value from its coset modulo $q$, we lift/decode to $\mathbb{Z}$ using the proper representatives. 

A:

My understanding is that the coefficients of polynomials used in lattice crypto are often sampled from a discrete Gaussian distribution.  A Gaussian is centered at 0, which would explain why the elements are represented as elements from the set $\{\frac{−(q−1)}{2},…,\frac{(q−1)}{2}\}$, as you mentioned.