Definition by Manetti (rephrased):$X$ is a topological space. $G \subseteq Homeo(X).$ Let $x,y \in X$. Then $x\sim y$ if $\exists g \in G$, such that $y=g(x)$. This defines an equivalence relation on X and whose quotient space is written as $X/G$.
I am confused about what is an element of $X/G$. For example, let $X = \mathbb{R}^n$ and $G=\{ f|f(x)=x+a, x \in \mathbb{R}^n, a \in \mathbb{Z}^n \}$ then what is in $X/G$?
On
I think the space would be characterized as the set of orbits by the action of $G$. More precisely, elements where you can move from $x$ to $y$ by a map in $G$.
In your example, $X/G$ are orbits of points by integer valued vector tranlation. Namely, if $x=(x_1,..,x_n)$ then its orbit will be $\{(y_1,...,y_n): x_i-y_i\in \mathbb{Z} \; \text{for all} \; 1\leq i\leq d \}$. You can also think of it as shifting all $\mathbb{Z}^n$ by $(\lfloor x_1 \rfloor,...,\lfloor x_n\rfloor)$. And as said in the answer by @Con, the space turns out to be $\mathbb{R}^n/\mathbb{Z}^n$.
In your example the space $X/G$ is just given by $\mathbb{R}^n/\mathbb{Z}^n$. To see that, let us check which kind of elements are identified. By definition, an element $x \in \mathbb{R}^n$ gets identified with all elements $y \in \mathbb{R}^n$ that arise as the image of $x$ under some element $g \in G$. But every element in $G$ maps $x$ to $x + a$ for some $a \in \mathbb{Z}^n$. That means the equivalence class of $x$ is exactly given by $x + \mathbb{Z}^n$. Hence the claim follows.