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.
Products $M^m \times S^{n-m}$ will be conformally flat, where $M^m$ is a compact manifold of curvature $-1$ and $S^{n-m}$ has curvature $1$. If $n>2m$ then the scalar curvature of the product will be positive (positive curvature of the sphere dominates the negative curvature of the hyperbolic manifold, so the scalar curvature of the product will be positive). You can also (frequently) deform these metrics a bit so that they no longer (locally) split as products and still have positive scalar curvature and remain conformally flat. Thus, you have counter-examples in all dimensions $n\ge 5$. See e.g. this paper by R.Mazzeo and N.Smale http://intlpress.com/JDG/archive/1991/34-3-581.pdf for further discussion.
Best Answer
All conformally flat homogeneous riemannian manifolds are symmetric spaces, by a result of Takagi. All homogeneous riemannian manifolds of dimension $\leq 11$ admit Einstein metrics, by results of Wang and Ziller. Most homogensous riemannian manifolds are not symmetric spaces.
References:
Takagi, H.: Conformally flat Riemannian manifolds admitting a transitive group of isometries I, II. Tohoku Math. J. 27, 103–110(I), 445–451(II) (1975)
McKenzie Y. Wang, Wolfgang Ziller: Existence and non-existence of homogeneous Einstein metrics, Inventiones mathematicae, 1986, Volume 84, Issue 1, pp 177-194