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 all precisely quotients of $\mathbb R^n$ by a discreet group of isometries, one should not expect to have a classification of such 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.
This is a very nice
The article of Kapovich is and from the its introduction to it 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, 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.