If a connected Riemannian manifold $M$ has an isometric transitive effective action of a Lie group $G$, then the manifold is diffeomorphic to $G/G_x$, where $G_x$ is the stabilizer of a point $x\in M$. Since $G_x$ preserves the metric at $T_x M$, then $G_x$ must be compact (the action $G_x:T_xM\to T_xM$ is faithful by considering the exponential map). Thus, one obtains all such manifolds by quotienting a Lie group by a compact subgroup. Another viewpoint is that the Lie algebra of the identity component of $G$ gives rise to a subspace of the Killing vector fields on $M$. When you restrict the vector space of Killing fields to a point $x$, then the image should map onto the tangent space $T_x M$. I think this is necessary and sufficient.

An interesting example is the Poincare dodecahedral space. In fact, one can decide which elliptic 3-manifolds are homogeneous, as the ones of the form $SU(2)/\Gamma$, where $\Gamma \leq SU(2)$ is finite.

If a connected complete (edit) Riemannian manifold $M$ has an isometric transitive effective action of a Lie group $G$, then the manifold is diffeomorphic to $G/G_x$, where $G_x$ is the stabilizer of a point $x\in M$. Since $G_x$ preserves the metric at $T_x M$, then $G_x$ must be compact (the action $G_x:T_xM\to T_xM$ is faithful by considering the exponential map). Thus, one obtains all such manifolds by quotienting a Lie group by a compact subgroup. Another viewpoint is that the Lie algebra of the identity component of $G$ gives rise to a subspace of the Killing vector fields on $M$. When you restrict the vector space of Killing fields to a point $x$, then the image should map onto the tangent space $T_x M$. I think this is necessary and sufficient.

An interesting example is the Poincare dodecahedral space. In fact, one can decide which elliptic 3-manifolds are homogeneous, as the ones of the form $SU(2)/\Gamma$, where $\Gamma \leq SU(2)$ is finite.