What is an example of an orbifold which is not a topological manifold?

It is quite easy to give an example in real dimension $4$.

In fact, it was shown by D. Mumford in the paper

The topology of normal singularities of an algebraic surface and a criterion for simplicity, Inst. Hautes Etudes Sci. Publ. Math. (1961), no. 9, 5 - 22

that a normal algebraic surface over $\mathbb{C}$, whose underlying topological space (in the usual topology) is a topological manifold, must be nonsingular.

Now take the orbifold $X:=\mathbb{C}^2/G$, where $G$ is the group of order $2$ whose generator $\xi$ acts as $\xi \cdot (x, \, y) = (-x, \, -y)$.

The $G$-invariant subalgebra is generated by $A:=x^2, \, B=xy, \, C:=y^2$, so the underlying algebraic surface $X$ is isomorphic to the affine quadric cone $\{B^2-AC=0 \} \subset \mathbb{C}^3$, which has a $A_1$-singularity at its vertex. Such a singularity is normal, because it is a codimension two hypersurface singularity.

Then $X$ cannot be a topological manifold by Mumford's result mentioned above.

To give a compact version of Igor's example, consider the antipodal map in the tangent space of a round 3-sphere at a point. This extends to an isometry of the 3-sphere with a pair of fixed points. The quotient by the isometry is a manifold with two conical singularities which is not homeomorphic to a topological 3-manifold.