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Here are two examples:

1) The pinched torus is not normal. It is a complex projective curve $C$ of equation $x^3+y^3=xyz$ in homogeneous coordinates $[x:y:z]$. It has a unique singular point $[0:0:1]$ and the link of this point $p$ is homeomorphic to two circles (we have $H_2(C,C-p;\mathbb{Z})\cong \mathbb{Z}\oplus \mathbb{Z}$).

2) The quadric cone is normal. It is an algebraic surface $S$ of equation $x^2+y^2+z^2=0$ in $\mathbb{P}^3(\mathbb{C})$ in homogeneous coordinates $[x:y:z:w]$ it has a unique singular point $[0:0:0:1]$. We notice that this space is homeomorphic to the Thom space of the tangent bundle of the $2$-sphere $S^2$. This remark gives a homeomorphism between the link of the singular point and the unit sphere bundle of the tangent bundle of $S^2$ which is connected (we get that $S$ is topologically normal).

Historicaly these two examples were important for our understanding of the failure of Poincaré duality for singular spaces, they appear in Zeeman's thesis:

E. C. Zeeman, "Dihomology III. A generalization of the Poincaré duality for manifolds",Proc. London Math. Soc. (3), 13 (1963), 155-183.

and also in McCrory's thesis:

C. McCrory, "Poincaré duality in spaces with singularities", Ph.D. thesis (Brandeis University, 1972)

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When you deal with statified pseudomanifolds

At least in the case of complex algebraic varieties one can give a nice topological interpretation of the normality condition.Let us consider $V$ a complex algebraic variety, you also have then its complex points $V(\mathbb{C})$ has the structure of a concept stratified pseudomanifold.

Let me recall that a stratified pseudomanifold $X$ is a filtered topological space$$X_0\subset\ldots \subset X_n$$such that each stratum, i.e. a connected component of $X_i-X_{i-1}$ is a manifold of dimension $i$ and such that $X_{n-1}=X_{n-2}$ and such that the regular part $X_n-X_{n-2}$ is dense in $X$. Together with a local condition: the existence of conical charts.

Thus $V(\mathbb{C})$ comes equipped with such a geometric structure. In the setting of stratified pseudomanifold one has a notion of normal pseudomanifoldspseudomanifold and normalization is a fundamental concept in intersection homology. A pseudomanifold $X$ of dimension $n$ is said to be normal if for every point $x\in X$ the local homology group $H_n(X,X-x,\mathbb{Z})$ is isomorphic to $\mathbb{Z}$. Now you know Notice that any complex algebraic variety has a structure of stratified pseudomanifoldhomological manifold is normal.

If you consider a triangulation $T$ of $X$ ($dim(X)=2$) $dim(X)=n$) then you can also prove that $X$ is normal if and only if the link of eack simplex in the $n-2$-skeleton of $T$ is connected. This is proved in Goresky, MacPherson "Intersection Homology theory" (Topology Vol. 19 (1980)). In this paper the authors also explains how to build normalization topologically and how topological normalization satisfies a universal property. In the case of $V(\mathbb{C})$ its topological normalization in the sense of Goresky-MacPherson is homeomorphic to $V'(\mathbb{C})$ where $V'$ is the algebraic normalization of $V$.

Thus topologically normality corresponds to the connectivity of the links, the link of a point in an $n$-dimensional manifold being a $n-1$ sphere we see that topological normalization is the very first step to desingularization of stratified pseudomanifolds.

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When you deal with statified pseudomanifolds, you also have a concept of normal pseudomanifolds. A pseudomanifold $X$ of dimension $n$ is said to be normal if for every point $x\in X$ the local homology group $H_n(X,X-x,\mathbb{Z})$ is isomorphic to $\mathbb{Z}$. Now you know that any complex algebraic variety has a structure of stratified pseudomanifold. Using Zariski’s Main Theorem, one can prove that a normal complex algebraic variety is a normal pseudomanifold.

If you consider a triangulation $T$ of $X$ ($dim(X)=2$) then you can also prove that $X$ is normal if and only if the link of eack simplex in the $n-2$-skeleton of $T$ is connected.

This is proved in Goresky, MacPherson "Intersection Homology theory" (Topology Vol. 19 (1980)).