As mentioned by @RyanBudney, homology between cycles is like a PL version of cobordism between manifolds inside an ambient space. Cobordism just like homotopy, but continuous deformation of one submanifold to another instead of just loops.
(note : I have added hover texts with the pictures, please place your mouse point over the pictures to see the corresponding texts explaining them)
$M, N$ be closed $n$-manifolds. $W$ be a $(n+1)$-manifold such that $\partial W = M \sqcup N$. Then $W$ is said to be a (unoriented) $\it{cobordism}$ between $M$ and $N$, and $M$ is said to be $\it{cobordant}$ to $N$. One can easily verify that cobordism is an equivalence relation: reflexivity ($M \sqcup M$ bounds $M \times I$) and symmetry is obvious, and transitivity can be seen by gluing cobordisms along boundaries.
Define $\Omega_n^{un}$ to be the collection of all cobordism classes of manifolds $[M]_{bor}$ with group operation $[M]_{bor} + [N]_{bor} = [M \sqcup N]_{bor}$. The identity under this group operation is the cobordism class $[\emptyset]_{bor}$ of the empty manifold. $\Omega_n^{un}$ is called the $n$-th unoriented cobordism group.
If $X$ is a simplicial/delta-complex, $\xi$ a simplicial $n$-cycle. Then $\xi$ can be thought, roughly, as a subcomplex of $X$ (by realizing each simplex in the formal linear combination as a simplex of the subcomplex, with products $n \cdot \Delta^i$ thought as an $i$-simplex thickened by a factor of $n$) with no boundary (i.e., no "open edges"). $\xi'$ be another $n$-cycle, realized as a subcomplex-ish thing.
If $\xi$ and $\xi'$ are homologous, the we know there is a $(n+1)$-chain $\sigma$ such that $\partial \sigma = \xi - \xi'$. But this $\sigma$ can also be realized as an $(n+1)$-dimensional subcomplex(-ish) of $X$ but with a few open edges this time such that boundary of the subcomplex is precisely $\xi - \xi'$. This reminds us of bordism, doesn't it?
Indeed, more formally, if we work in the singular context, then an $n$-cycle $\xi$ can be realized as a map $\xi : K_\xi \to X$ as follows: realize $\xi$ as the formal linear combination $\sum_i \varepsilon_i e^i$ where $e^i$ are singular $n$-simplices and $\epsilon_i$ are $\pm 1$ (repeatation is allowed in the sum). Glue $n$-simplices appropriately according to how pair of singular $(n-1)$-simplices cancel in $\partial \xi = 0$ (there is a little subtlety here, but let's ignore that for now). We get the delta complex $K_\xi$ and the map $K_\xi \to X$ is obtained from sending each simplex to $X$ by the singular simplices $e^i : \Delta^i \to X$.
Thus, singular $n$-cycles are merely maps $\Delta \to X$ from delta complexes. Given two such maps $\xi : \Delta^n \to X$ an $\xi' : \Delta'^n \to X$, we see $\xi$ and $\xi'$ are homologous if $\xi$ and $\xi'$ extend to a map $\zeta : \mathbf{\Delta}^{n+1} \to X$ where $\partial \mathbf{\Delta}^{n+1} = \Delta^n \sqcup \Delta'^n$, i.e., $\zeta|_{\Delta^n} = \xi$ and $\zeta|_{\Delta'^n} = \xi'$. One can similarly check that this is an equivalence relation, and the equivalence classes of maps from certain delta complexes to $X$ forms precisely the $n$th singular homology group.
So homology between cycles in $X$ is merely (sort-of-)cobordism between simplicial complexes inside the ambient space $X$.
