I have two basic and simple question. I need to understand it.
(i) Why and how cohomology is the dual concept of homology ? Can you attach an example showing this ?
(ii) Why the word "singular" is attached to the definition of homology or cohomology?
The answer to question (1) is "yes, but not in a natural nor naive way". What I mean is that you cannot expect something like $H^n(X)\simeq \hom (H_n(X),\mathbb Z$ in general.
Relationships between cohomology and dual of homology are established by the Universal coefficients Theorem. Here's a version for abelian groups.
Let $X$ be a topological space and let $A$ be an abelian group. Then let $H_n(n,A),H^n(X,A)$ respectively be the $n$-th homology and cohomology groups with coefficients in $A$.
Then there is an exact sequence $$ 0\to \mathrm{Ext}^1_{A}(H_{n-1}(X,\mathbb Z),A)\to H^{n}(X,A)\to \\ \to\hom(H_{n}(X,A),A)\to 0 $$
moreover, this sequence splits (not canonically) so
$$H^n(X,A)\simeq \mathrm{Ext}^1(H_{n-1}(X,\mathbb Z),A)\oplus \hom( H_n(X,A),A)$$
The summand $\mathrm{Ext}^1(H_{n-1}(X,\mathbb Z),A)$ does not vanish in general, even if $A=\mathbb Z$ (which computes singular cohomology) as
$$\mathrm{Ext}^1(H_{n-1}(X,\mathbb Z),\mathbb Z)\simeq \mathrm{tors}(H_{n-1}(X,\mathbb Z))$$
while if $H_n(X,\mathbb Z)$ is torsion, then $\hom(H_n(X,\mathbb Z),\mathbb Z)=0$.
For the second question, the adjective "singular" is used to denote how the (co)cycle complex was obtained. The point is that, in singular cohomology, one considers $C^n$ as free abelian group generated by continuous maps $\sigma :\Delta ^n\to X$ where $\Delta ^n$ is the usual simplex; the image of $\sigma$ can be thought as a $n$-simplex "with singularities".
This paper gives what seems to me useful historical references and also a discussion of the oriented as against ordered case.
