See definition and examples here. An abstract simplicial complex $K$ is just a set of points and subsets of these points satisfying some simple conditions. Its topological realization, denoted $|K|$, is a corresponding Topological space gotten from the disjoint union of simplices associated with these subsets, quotient certain inclusion relations (similar but not the same as Delta-complexes).

We also define a Triangulation of a space, as well as Simplicial maps between simplicial complexes.

A more flexible way of constructing topological spaces are CW complexes.

Can be used to calculate Fundamental groups

Traditionally, simplicial homology is defined for simplicial complexes, which are the ∆ complexes whose simplices are uniquely determined by their vertices. This amounts to saying that each n simplex has n + 1 distinct vertices, and that no other n simplex has this same set of vertices.