Suppose one relaxes the condition that the action is free with the condition that every point in $M$ has a finite $G$-stabilizer. In this case, the topological quotient $M/G$ carries a natural orbifold structure. The local model is therefore the quotient of $\mathbb{R}^n$ by a finite group acting linearly.
In general, $M/G$ is a stratified space, and the stratification is indexed by conjugacy classes of $G$-stabilizers.