locally conformally flat manifold

The simplest example is $S^n$, it is locally conformally flat with the standard metric, and is not flat for obvious reasons.

While flat manifolds are precisely quotients of $\mathbb R^n$ by discreet group of isometries, one should not expect to have a classification of conformally flat manifolds in higher dimensions. For example, already in dimension $4$ it was proven by Kapovich in

M. Kapovich. Conformally flat metrics on 4-manifolds. J. Differential Geom. 66 (2004), no. 2, 289–301,

that arbitrary finitely presented group can be a subgroup of a fundamental group of a conformally flat manifold.

The article of Kapovich is and from its introduction you will learn a lot on the question. $4$-dimensional manifolds with LCF structure have zero signature, in dimension $3$ it is known that some manfiolds don't admit conformally flat structure, first example was constructed in W. Goldman, Conformally flat manifolds with nilpotent holonomy and the uniformization problem for $3$-manifolds, Transactions of AMS 278 (1983).

One more remark -- all hyperbolic manifolds (of constant negative sectional curvature) are all conformally flat. A connected sum of two conformally flat manifolds is conformally flat and so this already gives you a large collection of examples.

A simple obstruction is this: No compact (without boundary), simply-connected $n$-manifold that is not diffeomorphic to the $n$-sphere carries a locally conformally flat structure. The reason is that the developing map construction shows that any locally conformally flat metric $g$ on a simply-connected $M^n$ is, up to a conformal factor, the pullback of the standard metric on $S^n$ under some immersion $\phi:M^n\to S^n$. If $M$ is also compact without boundary, then $\phi$ is a covering map and, hence, a diffeomorphism.