Importance of triangulation Kervaire's seminal 1960 paper A manifold which does not admit any Differentiable Structure starts "An example of a triangulable closed manifold $M_0$ of dimension 10 will be constructed."
What is the importance of triangulability of a manifold? It suggests a certain degree of "niceness" but what is gained by this assumption precisely? 
 A: I don't think you should view a triangulation as a "niceness" feature of a manifold.  Keep in mind that this paper was one in a chain where people were answering various foundational issues that came-up when dealing with the concept of a manifold. 
It goes back to Poincare's work.  His proof of the duality theorem for manifolds assumes manifolds are triangulated.  In particular, his definition of homology required a triangulation.  So a natural question people came to ask was if every manifold has a triangulation, and whether or not they're essentially unique (up to subdivisions).  Other side-questions were whether or not you could define homology without triangulations, in a homotopy-invariant way, etc. 
Poincare assumed manifolds were smooth, and at that level of generality the question was answered by Whitehead: smooth manifolds admit unique smoothly-compatible triangulations up to subdivision.   But topologists eventually took this foundational problem in a more general light.  With the development of point-set topology one can ask about triangulations of topological manifolds.  Here things become more subtle and technical, as it is asking for a feature which is "native" to smooth manifolds (triangulations) to apply to a very different kind of object, a non-smoothable manifold. 
So this is much like trying to find analogues of ideas from Riemann manifolds for plain metric spaces (say, metric spaces with path-metrics).  
