This article says the following:
Let $X$ be a triangulated space and let $C_n(X)$ be a real vector space with $n$-simplices $[x_0,x_1,x_2,\dots,x_n]$. Each different combination of $x_i's$ forms a different basis for our vector space.
Let $X$ be $I^3$ (which is $[0,1]\times[0,1]\times[0,1]$). This is clearly a tringulated space. What is $C_3(I^3)$?
Moreover, what is the basis of $C_3(I^3)$?
A triangulated space means: a space (which admits at least one triangulation) together with a fixed triangulation of it. This means, you consider this fixed triangulation, which gives you e.g. a finite set of $n$-vertices, which become a basis of $C_n(X)$ after ordering them.
Similar situation: Oriented manifold or orientable manifold. In the latter we just assume existence of an orientaion, in the former we want to come the space together with a preferred orientation.