Configuration spaces refer to topological spaces that consist of ordered or unordered subsets of a topological space, of a given (finite) cardinality.
Configuration Spaces typically refer to $C_n X = X^n \setminus \Delta$ where $\Delta$ refers to the subset of $X^n$ where any two coordinates agree,
$$\Delta^n = \{ (x_1, \cdots, x_n) \in X^n : x_i = x_j \text{ for some } i \neq j\}$$
$X$ is a topological space.
There is a natural right action of the symmetric group $\Sigma_n$ on $C_n X$. For example, $\pi_1 C_n D^2$ is the pure braid group on $n$ strands, and $\pi_1 (C_n D^2 / \Sigma_n)$ is the braid group on $n$ strands.
