What are $U(n)$ or $\mathbb{Z}_2$ quantum spin liquids? Quantum spin liquid is a state of matter in which spins are correlated and fluctuate even at zero temperature.
My question is about these terms in general. When we say that a state or a quasi-particle is $U(n)$, $SU(n)$ or $\mathbb{Z}_2$, what do we physically mean? 
 A: They specify which gauge symmetry the Quantum Spin Liquids is subject to, when treated via a Lattice Gauge Theory. The gauge field (some sort of interaction) is defined over a discretised space(-time).
For example imagine atoms are situated on the vertices on this square lattice:

where the lines between vertices are links, and the area defined by $4$ links is a plaquette.

$U(1)$
Gauge variables are unphysical and not directly observable. Let's start with a $U(1)$ gauge theory, corresponding to usual electromagnetism.
The gauge fields live on links like $nm$. Physical (observable) variables are electric fluxes on links $E_{nm}$ and magnetic fluxes through plaquettes $\Phi_{mnpq}$:
$$ \mathbf{E} = -\frac{\partial \mathbf{A}}{\partial t} \quad\Rightarrow \quad E_{mn} = -\dot{A}_{mn}, $$
$$ \Phi = \int \mathbf{B}\cdot \mathrm{d}^2\mathbf{r} = \oint_{\mathrm{plaquette}} \mathbf{A}\cdot \mathrm{d}\mathbf{r} \quad \Rightarrow \quad \Phi_{mnpq} = A_{mn}+A_{nq}+A_{pq}+A_{qn}.$$
$E$ is quantised, because it is related to electric charges. In some units, then, $E = 0, \pm 1, \pm 2, \dots$.
Normally $A$ can take any value from $-\infty$ to $\infty$. Usually, however, one considers a compact U(1) gauge theory by requiring $$ 0 \leq A \leq 2\pi, \quad V(A + 2\pi)  = V(A) \Rightarrow V(A) = f(e^{\mathrm{i}A}), $$
where $V$ would be the potential energy in the Hamiltonian $H$:
$$ H = \sum_{\mathrm{links}} \frac{E^2_{mn}}{2I} - \sum_{\mathrm{plaquette}} \lambda \cos \Phi_{mnpq}.$$
The $E^2$ term makes sense as it is the energy density of the electric field. The $\cos$ is introduced to preserve the compactness and periodicity of the magnetic potential $A$. For small fluxes $\Phi$, this reduces to $\propto 1 + B^2$ which, again, makes sense as it is the energy density of the magnetic field.

$\mathbb{Z}_2$
For $\mathbb{Z}_2$, you switch from integer arithmetics $(\mathbb{Z})$ to binary arithmetics $(\mathbb{Z}_2)$ for the electric flux along the links. They can be either "up" or "down", and not taking any value $\in \mathbb{R}$ between $0$ and $2\pi$.
The simplest way to try and do this is to exponentiate all your old variables $E$, $A$ and $\Phi$:
$$ \begin{array}{|c|c|}
\hline
\mathrm{(compact)}\, U(1) & \mathbb{Z}_2 \\
\hline
E = 0, \pm 1, \pm 2, \dots & (-1)^E \leftrightarrow \sigma^x = \pm 1 \\
0 \leq A \leq 2\pi & e^{\pm \mathrm{i}A} \leftrightarrow \sigma^z = \pm 1 \\
\Phi = \sum_{\mathrm{plaquette}}A & e^{\mathrm{i}\Phi} \leftrightarrow \Pi_{\mathrm{plaquettes}} \sigma^z = \pm 1 \\
H = \sum_{\mathrm{links}} \frac{E^2_{mn}}{2I} - \lambda\sum_{\mathrm{plaquette}} \cos \Phi_{mnpq} & H = -\Gamma\sum_{\mathrm{links}}\sigma^x_{mn} - \lambda \sum_{\mathrm{plaquettes}} \sigma^z_{mn}\dots \sigma^z_{qm}\\
\hline
\end{array}$$
An example of a $\mathbb{Z}_2$ theory, with $\sigma_x$ and $\sigma_z$, is the Heisenberg model. This can employed, for instance, to model the Kagome lattice -- which is one of those expected to display quantum spin liquids, and which hopefully someone will soon investigate experimentally!:


EDIT: reference for figures: here and here.

$U(N)$, $SU(N$) etc.
If you want to have $SU(2)$, instead of $U(1)$, you need a state which has $2$ internal degrees of freedom (internal states). So you don't just have a scalar wavefunction $\psi \rightarrow \underbrace{e^\phi}_{\in U(1)}\psi$, but a multi-component one $\psi = 
\left ( \begin{array}{c}
\psi_1\\
 \psi_2
\end{array}
\right ) \rightarrow \underbrace{\left ( \begin{array}{cc} 
a & b \\
c & d 
\end{array} \right )}_{\in SU(2)} \left ( \begin{array}{c}
\psi_1\\
 \psi_2
\end{array}
\right ) $, and so on for $ N \geq 2$.
