From the comments: 

1. Suppose that $T$ is a triangulation.  If all vertex links are PL $(n-1)$-dimensional spheres then the realisation space of $T$ is a PL manifold and thus a topological manifold. (In the compact case this is equivalent to the usual definition.)

2. There are triangulations of topological manifolds (in fact, of $S^5$) that do not have this property. Examples come from the [double suspension theorem][1] of Cannon and also Edwards. 

Furthermore: 

3. There are triangulations of topological manifolds that have no PL structure. This is (essentially) the failure of the [Hauptvermutung][2].

4. There are closed topological manifolds that do not admit any triangulation.  This is the failure of the Triangulation Conjecture, due to Casson in dimension four and to [Manolescu][3] in all higher dimensions.


  [1]: https://en.wikipedia.org/wiki/Double_suspension_theorem
  [2]: https://en.wikipedia.org/wiki/Hauptvermutung
  [3]: https://arxiv.org/abs/1607.08163