From Hatcher, "Algebraic Topology," Chapter 2, "Singular Homology" section (p. 108-109 in my copy):
Cycles in singular homology are defined algebraically, but they can be given a somewhat more geometric interpretation in terms of maps from finite $\Delta$-complexes. To see this, note that a singular $n$-chain $\xi$ can always be written in the form $\sum_i \varepsilon_i \sigma_i$ with $\varepsilon_i = \pm 1$, allowing repetitions of the singular $n$-simplices $\sigma_i$. Given such an $n$-chain $\xi = \sum_i \varepsilon_i \sigma_i$, when we compute $\partial \xi$ as a sum of singular $(n-1)$-simplices with signs $\pm 1$, there may be some canceling pairs consisting of two identical singular $(n-1)$-simplices with opposite signs. Choosing a maximal collection of canceling pairs, construct an $n$-dimensional $\Delta$ complex $K_\xi$ from a disjoint union of $n$-simplices, one for each $\sigma_i$, by identifying the pairs of $(n-1)$-dimensional faces corresponding to the chosen canceling pairs. The $\sigma_i$'s then induce a map $K_\xi \rightarrow X$. If $\xi$ is a cycle, all the $(n-1)$ simplices of $K_\xi$ come from canceling pairs, hence are faces of exactly two $n$-simplices of $K_\xi$. Thus $K_\xi$ is a manifold, locally homeomorphic to $\mathbb{R}^n$, except at a subcomplex of dimension at most $n - 2$.
It's this last bit that has me confused, perhaps because I'm unable to visualize a nontrivial 3-manifold. We're building a $\Delta$-complex by identifying the $(n-1)$-faces of simplices; there are only finitely many such simplices; we're not required to identify the faces in any unusual way; and so forth. How, then, can we end up with something which is not a manifold? I'd really appreciate an explicit example.
I've been told that this is a general example of a "pseudomanifold," but all the examples of pseudomanifolds that I've been able to locate and follow wind up making a non-manifold by essentially identifying vertices to get pinched spaces. This can't happen under the present construction because we're always identifying the largest proper faces. So I'm quite confused by the situation.
EDIT: Looks like there are some more questions raised in the comments as to what the construction actually means. If there's a name for this construction, or any other source that discusses it, I'd appreciate a reference. And of course if anyone can shed any additional light on the topic that would be quite welcome too.
EDIT: I may have inadvertently overwritten someone else's edit just then, judging by a message that popped up. Apologies if so. (How do I tell / fix this?)