I am reading singular homology in Hatcher. Let $X$ be a topological space. We let $C_n(X)$ be generated by singular $n$-simplicies $\sigma:\Delta^n\to X$.
What I don't understand is, what collection of $n$-simplicies do we want? Do we just take every single possible distinct map $\Delta^n\to X$? Say I am mapping $3$-simplicies into the space $X=\Delta^3$, do I not have uncountably many choices of maps?
I can see in this way why the point space has a single singular $n$-cell for each $n$, since we must map each simplex to the point, if we are taking every possible map.
Historically homology groups were first defined for simplicial complexes which are purely combinatorial entities. See e.g. https://en.wikipedia.org/wiki/Simplicial_complex and https://en.wikipedia.org/wiki/Simplicial_homology.
Let us consider your example $\Delta^3$. We obtain one generator of the non-oriented simplicial chain complex in dimension $3$ resp. two generators of the oriented simplicial chain complex in dimension $3$. This is a very effective method to associate chain complexes and homology groups to simplicial complexes.
What is the relationship between topological spaces and simplicial complexes? Each simplicial complex $K$ has a geometric realization $\lvert K \rvert$ which is a topological space consisting of Euclidean simplices which are convex hulls of points in general position in some $\mathbb{R}^n$ (each combinatorial $n$-simplex which is a finite set of vertices is realized as a Euclidean simplex, and these Euclidean simplices are glued together by the corresponding combinatorial incidences of the faces). But certainly not every space is the geometric realization of simplicial complex. Moreover, two non-isomorphic simplicial complexes may have homeomorphic geometric realizations - are their homology groups isomorphic ("topological invariance of simplicial homology")? It was discovered that this question cannot be answered by purely combinatorial methods. This was the origin of singular homology. A singular simplex in a space $X$ is any map $\sigma : \Delta^n \to X$. In general this produces uncountably many singular simplices (even for the simplest spaces). The resulting chain complexes are very big, but if you go to homology groups the size massively collapses. In fact, the singular homology groups of the geometric realization $\lvert K \rvert $ of a simplicial complex $K$ are isomorphic to the simplicial homology groups of $K$. This proves the topological invariance of simplicial homology and provides a method to effectively compute the homology groups of polyhedra (spaces $X$ which have triangulations, i.e. admit a homeomorphism $h ; X \to \lvert K \rvert$ where $K$ is simplicial complex).
Singular homology is excellent because it is conceptually simple. It is defined for any space without referring to additional structural components (which are usually not uniquely determined by the space itself, see e.g. https://en.wikipedia.org/wiki/Hauptvermutung), even the beginner will easily understand it and topological invariance is obvious from the definition (that is why we take all singular simplices and don't choose a "smaller" assortment). However, it is practically impossible to compute singular homology groups based on the definition of the singular chain complex. But singular homology satisfies the Eilenberg–Steenrod axioms https://en.wikipedia.org/wiki/Eilenberg%E2%80%93Steenrod_axioms and we can compute a lot of homology groups using only these axioms. See any book on algebraic topology. The Eilenberg–Steenrod axioms determine homology theories on finite CW-complexes uniquely (up to natural isomorphism) by their coefficient groups $H_0(\ast)$ (where $\ast$ is a one-point space). In case of the standard singular homology theory we have $H_0(\ast) \approx \mathbb{Z}$ which is one of the few cases where an effective computation is possible.
My opinion is that the main purpose of singular homology is to establish the existence of a homology theory which satisfies the Eilenberg–Steenrod axioms and is defined for all topological spaces. There are a lot of other constructions, but they are not simpler and some are only defined on certain classes of spaces. The material distinctions between the various constructions are unveiled only on spaces which are no finite CW-cmplexes, and then it becomes apparent that there are homology theories which perform better then singular homology.