What is an example of an orbifold which is not a topological manifold? In Thurston's book The Geometry and Topology of Three-Manifolds it is proven that the underlying space of a two-dimensional orbifold is always a topological surface. 
Are there any easy examples of higher dimensional orbifolds whose underlying spaces are not topological manifolds?
 A: 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.
A: 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.  
