To answer the question *What are quantum integrable systems ?* let us find common understanding what is quantization. Roughly speaking it is the following - you have some classical phase space e.g. $\mathbb{R}^{2n}$ with coordinates $q_i$ and $p_i$. Now corresponding **quantum algebra** of observables is an algebra with generators $\hat p_i, \hat q_i$ such that commutators $[\hat p_i , \hat q_j]=\delta_{ij}$.

Now assume you have classical **hamiltonian** integrable system, so you have functions $H_i(p,q)$ such that they Poisson commute.

The first aim of quantization of integrable system is to find elements $\hat H_i( \hat p , \hat q)$,
such that their commutator will be equal to zero and their principal symbols equals to classical $H_i$.

**Answer 1a:**
So a quantum integrable system corresponding to classical hamiltonian system defined by $H_i$ is
given by $\hat H_i$, with properties above.

In general you may not consider $\mathbb{R}^{2n}$, but some symplectic manifold, and deformation quantization of algebra of functions on it. Quantization of integrable system - is again looking for $\hat H_i$ in quantum algebra corresponding to $H_i$ in algebra of functions on symplectic manifold.

Remark: Well, may you consider Poisson manifolds as well, and classical integrable systems are maximal Poisson commutative subalgebras and we may look for they lift to quantum algebras. Actually many practical examples work with Poisson manifolds g^*, for Lie algebras $\frak g$.

To the best of my knowledge the question: `Is it always possible to do so or not?', is open in general.
And in many case it is an **art** to find corresponding $H_i$. Some recent research on general question can be found here. Analysis of many concrete examples is related to representation theory, quantum groups etc...

**Answer 1b:** Are there examples that are not too complicated?

Trivial example - take $\mathbb R^{2n}$, take $H_i = p_i$, (free motion) corresponding quantum system $\hat H_i = \hat p_i$.

One more trivial example $H_l=p_l^2 + q_l^2$ (independent Harmonic oscillators), corresponding $\hat H_i = \hat p_i ^2 + \hat q_i^2$.

Let me give non-trivial examples later - you can search for Calogero system, Toda, Gaudin model.

**Answer 2:**

Why are mathematicians interested in quantum integrable systems?

1) In many cases study of quantum integrable systems is the same as certain representation theory questions - e.g. corresponding $\hat H_i$ often have a sense of centres of universal enveloping or their quantum deformations MO question about the Yangian

2) It is related to certain "hot topics" as Langlands correspondence (e.g. http://arxiv.org/abs/hep-th/0604128 ), quantum cohomology ( e.g. http://arxiv.org/abs/1211.1287 )

3) We might want to apply math. to something "physically" sounding :)