The Wikipedia article defines a non-linear sigma model as model for a scalar field $\Sigma$ which takes on values in a nonlinear manifold called the target manifold $T$.
What is the definition of a nonlinear manifold? I am wondering if that means a curved manifold, but in that case what precisely is the non linearity?
It is my understanding the term non-linear manifold simply refers to the target space of non-linear $\sigma$-models in the literature. To be explicit, consider for example the $O(3)$ $\sigma$-model,
$$S = g^2 \int \partial_\mu n^a \partial^\mu n_a + i \theta \, H(n^a)$$
where $n^a$ is a unit vector field ($n^a n_a=1$) in $2+1$ dimensional space and $H$ is the Hopf invariant. At any time, for finite energy solutions, one has the boundary condition, $n \to (0, 0, 1)$.
In this theory, space-time may be identified with $CP^1$ (or equivalently $\mathbb C \cup\{\infty\}$) and the configuration space consists of maps $S^2 \to S^2$. The manifold is referred to as a non-linear manifold in this context.
More generally, a non-linear $\sigma$-model may be written as,
$$S=\int_\Sigma g_{ab}(\phi) \partial^\mu \phi^a\partial_\mu \phi^b$$
where $\phi^a$ map from $\Sigma$ to some target space $M$. Notice the 'non-linear' part is that the metric tensor is allowed to be a function of the fields themselves, and is contracted with the fields.
Essentially, we can write any non-linear $\sigma$-model in such a way that the collection of fields $\phi^a(x^\mu)$ can be viewed as embedding functions of some manifold $\Sigma$ in $M$.
For example, for the $O(3)$ model, solving the constraints in terms of some parameter (interpreted as a field) and substituting it back into the action will cast it in this form.
What is meant is that the symmetry group $G$ does not act linearly on the fields, i.e., they do not live in a vector space (linear) representation of the symmetry group $G$. For example take the Lagrangian $$L = \frac{m}{2}\big[\dot{r}^2 + r^2 (\dot{\theta}^2 + (\sin^2\theta)\dot{\varphi}^2) \big] \tag 1 $$ which has a non-linear $U(1)$ invariance $$\varphi \overset{g}{\mapsto} \varphi + g.$$ where $\varphi$ takes values not on the real line but on the circle. The circle happens to be the same as $U(1)$, but of course $G$ can also act on a different manifold than itself. For example, (1) is also invariant under $SO(3)$ rotations with $(\theta,\varphi)$ coordinates on the 2-sphere.
I am not sure if it's useful to think in terms of curvature. The Lagrangian $L$ is of course $\propto g_{ab} \dot{x}^a \dot{x}^b$ for $g_{ab}$ the flat Euclidean metric, just expressed in spherical coordinates. And a 1-dimensional manifold is always flat, so if you have several commuting symmetries you can have a flat torus.
Weinberg, Vol. 2, Ch. 19, esp. Sec. 19.6 discusses this topic in much detail.
