Is every quotient by a finite group an orbifold? It is required, in order to be an orbifold, to be locally like $\mathbb{R}^n/\Gamma$ where $\Gamma$ is a finite subgroup of $GL(n,\mathbb{R})$ and that the fixed points of the action of $\Gamma$ have codimension at least $2$.

I wonder if the quotient of a manifold $M$ by the action of a finite group $G$ is always an orbifold, or under which condition it is.

I know that by the slice theorem, since $G$ is compact $\dim(G) = 0$ exists a tubular neighborhood of any orbit $G\cdot x$ that is $G$-invariant. In particular we can identify this with a finite number of open balls of $\mathbb{R}^n$ but this is not enough to prove all the points of the definition (we want a representation of $\Gamma$ on $GL$ and thus embed the tubular neighborhood into $\mathbb{R}^n$ in an equivariant way). 
This question is motivated by the following example: consider 
$$M^N = M\times M\times\dots \times M$$
n-fold product and the action of the symmetric group $S_N$ on it by permutations, 

will $M^N/S_N$ be  and orbifold?

 A: There is a stronger version of the equivariant tubular neighborhood theorem, which gives the essentially affirmative answer to your question. It is worth knowing in any case. (I do not know a reference off-hand, but I would be surprised if whatever reference you were looking at did not really prove this.) 

Let $M$ be a smooth $G$-manifold, with $G$ a compact Lie group. If $S$ is a closed, $G$-invariant submanifold of $M$, then the bundle $NS$ carries a $G$-action by bundle isomorphisms covering the action on $S$; there is a $G$-invariant neighborhood of $S$ which is equivariantly diffeomorphic to a neighborhood of the $0$-section in $NS$.

Given any point $x \in M$, there is thus an orbit $Gx$ through $x$, diffeomorphic to $G/\Gamma_x$, where $\Gamma_x$ is the stabilizer, and this is the smallest $G$-invariant submanifold containing $x$. A $G$-vector bundle over $G/\Gamma_x$ is precisely the data of a $\Gamma_x$-representation $V$; the corresponding vector bundle is $G \times_{\Gamma_x} V$. 
When $G$ is a finite group, this says that at a point $x \in M$, there is a $\Gamma_x$-invariant neighborhood $U_x$ of $x$ which is diffeomorphic to a neighborhood of $0$ of some $\Gamma_x$-representation $V$; the corresponding neighborhood of $Gx$ is $$\bigoplus_{[g] \in G/\Gamma_x} U_{gx},$$ and taking the quotient by $G$ gives $U_x/\Gamma_x$. As above, this is a neighborhood of zero in the quotient of a $\Gamma_x$-representation. 

I am not familiar with the restriction you make in your definition of orbifolds (I don't know much about orbifolds), but because one recovers the $\Gamma_x$-invariant neighborhood of $x$ in terms of the $\Gamma_x$ action on $T_x M$, observe that this is equivalent to saying that $M/G$ is an orbifold iff $T_x M^{\Gamma_x}$ has codimension at least $2$ for all $x$. (What we should probably say is that $(T_x M)^{\Gamma_x}$ has codimension at least $2$ so long as $\Gamma_x$ is nontrivial.) 

$M^N/\Sigma_N$ will only satisfy your condition if $\dim M > 1$, but then it will always hold.
Write a point $x = (x_1, \cdots, x_N)$ in the manifold $M^N$. Suppose for convenience that this is $$x_1 = \cdots = x_{n_1},$$
$$x_{n_1+1} = \cdots = x_{n_2}, $$ and so on until $x_{n_k} = x_N$, and otherwise no $x_i$ is equal. If we write $\ell_i = n_i - n_{i-1}$ (interpreting $n_0 = 0$), then the stabilizer of $x$ is the product of symmetric groups $$\Gamma_x = \Sigma_{\ell_1} \times \cdots \times \Sigma_{\ell_k},$$ where  $\ell_1 + \cdots + \ell_k = N$. The action on $T_x M^N = (T_{x_i} M)^N = T_x M \otimes \Bbb R^N$ is induces by the action of this group on $\Bbb R^N$ (thinking of it as a subgroup of $\Sigma_N$). 
The action of this group on $T_x M^N = \oplus_{i=1}^k (T_{x_{n_i}})^{\ell_i}$ is by the action of $\Sigma_{\ell_i}$ on each factor by permutations. Furthermore, the fixed point set of $\Sigma_j$ on $\Bbb R^j$ is 1-dimensional for all $j$; so the fixed point set of the action of the $\Gamma_x$ on $T_x M^N$ is $\oplus_{i=1}^k T_{x_{n_i}}$, and in particular of dimension $k\dim M$, where $k$ is the number of terms in some partition $\ell_i$ of $N$; the codimension is $(N - k)\dim M$. In particular, if the codimension is $0$ then $\Gamma_x = 1$ (none of the $x_i$ are equal); and the minimal codimension is $\dim M$. This gives the desired result so long as $\dim M > 1$. 
(As a topological space, $\text{Sym}^2(S^1)$ is the Mobius band; as an 'orbifold', the points on the manifold boundary have orbifold-y charts modeled on $\Bbb R^2/\Sigma_2$ with $\Sigma_2$ acting by the coordinate-swap map.)
