What is the point of triangulating topological spaces? In a general sense, what is the purpose to triangulating, for example, a 3-dimensional topological space? What advantages does it give if we can triangulate a Seifert-Weber space into 23 tetrahedra?
Wikipedia says homology and cohomology groups can be computed with simplicial homology and cohomology theories, as opposed to more complicated homology and cohomology theories.
I don't really understand this as my grasp of topology and geometry is quite weak. I would appreciate it if someone could answer this in a simple manner (examples of where triangulations are useful in real life applications would be awesome!!) 

If you only know about singular homology, which is defined directly for topological spaces, most explicit calculations are very hard. Having a triangulation as a $\Delta$-complex, simplicial-complex or even just a CW-complex reduces the topological space to a finite set of gluing data of simplices (or spheres in the CW-case). This data can be put into a computer and the computer can tell you all about the homology and cohomology groups of the complex. Then you have theorems telling you that singular homology and say simplicial homology are isomorphic, so you really learned something about your triangulated space! So the main advantage of triangulations in algebraic topology is being able to explicitly compute a lot of algebraic invariants of a topological space.
