In the context of Goldman's paper *The symplectic nature of fundamental groups of surfaces*:


Consider a closed oriented surface $S$ with fundamental group $\pi$, and let $G$ be a connected Lie group. The space $\operatorname{Hom}(\pi,G)$ consisting of representations $\pi\to G$ is ~~a real analytic variety~~ a real analytic space (possibly singular). There is a canonical $G$-action on $\operatorname{Hom}(\pi,G)$ obtained by composing representations with inner automorphisms of $G$. When is the representation space $\operatorname{Hom}(\pi,G)/G$ a smooth manifold? I think that if the representation is irreducible, the space is smooth manifold nearby.



Space of representations of surface group into Lie groups
The question of the structure of $Hom(\pi, G)$ and $Hom(\pi, G)/G$ is very complicated for a general connected Lie group $G$, though less so for a surface group than for a general finitely generated group. To see just how wild things can get, you should look at this paper of Kapovich and Millson <http://www.math.umd.edu/~millson/papers/char19.pdf>.


For a surface $\Sigma$, if $G$ is Abelian, then 


$Hom(\pi, G)\simeq Hom(\pi, G)/G\simeq H^{1}(\Sigma, G)\simeq G^{2\sigma}$


where $\sigma$ is the genus of $\Sigma.$ In particular, in this case, it is naturally a smooth, even real analytic, manifold in a canonical way.


Meanwhile, as mentioned in the above comments, there are situations when $Hom(\pi, G)$ and/or $Hom(\pi, G)/G$ is homeomorphic to a manifold, although this is something of an accident.


Probably the cleanest statement available that lives in some level of generality, is that if $G$ is a connected reductive complex affine algebraic group (like $GL\_{n}(\mathbb{C})$), then 
$Hom(\pi, G)$ is a complex affine variety with at most quadratic singularities, this is due to Goldman, Millson and Simpson (<https://projecteuclid.org/euclid.bams/1183554530>), and remarkably this result remains true as long as $\pi$ is the fundamental group of a compact Kahler manifold. But, even in this case, the quotient $Hom(\pi, G)/G$ is a total mess, in particular it is Hausdorff only when $G$ is Abelian.


This can be remedied (to some extent) in this case by studying the GIT quotient $Hom(\pi, G)//G,$ but since you ask about real Lie groups in general, GIT theory again is a bit of a mess since $\mathbb{R}$ is not algebraically closed. 


As mentioned in the comments, if you take $G$ a connected semi-simple real Lie group, then the subset $Hom^{\star}(\pi, G)$ consisting of irreducible representations whose centralizer is equal to the center of $G,$ then this is a smooth (even real analytic) manifold upon which $G$ acts with constant stabilizer equal to the center, and properly, and therefore $Hom^{\star}(\pi, G)/G$ is a (real analytic) manifold.


Just as a final remark, there are many very good reasons why one might consider the space of conjugacy classes of all representations $\pi\rightarrow G,$ but as explained above, this is basically hopeless in any reasonable category of classical spaces such as: smooth manifolds, algebraic varieties, real analytic spaces...


One remedy to this hopelessness is to pass to a suitable category of stacks, but of course it heavily depends on what you want to do. 


