a simplicial complex
Build a shape out of dots, sticks, solid triangles, solid tetrahedra, and their higher cousins, glued together only along whole shared faces. That is a simplicial complex: a Lego model of a space where the bricks are simplices — a 0-simplex is a vertex, a 1-simplex an edge, a 2-simplex a filled triangle, a 3-simplex a filled tetrahedron — and the gluing rules are strict, so two triangles may share an entire edge or a single vertex but never overlap halfway or cross through each other's interiors.
There are two flavors that you should keep apart. A geometric simplicial complex K is an actual collection of simplices sitting in R^N such that (i) every face of a simplex in K is again in K, and (ii) the intersection of any two simplices is either empty or a common face of both. An abstract simplicial complex throws away the geometry: it is just a set V of vertices together with a family of finite nonempty subsets (the simplices) that is closed under taking nonempty subsets — if {a, b, c} is a simplex then so are {a, b}, {a}, and so on. Every abstract complex can be realized geometrically (its geometric realization |K|), so the abstract version is the bookkeeping and the geometric version is the picture. Computing topology then becomes finite combinatorics: the homology H_k(K) is read off the boundary operators between the chain groups spanned by the k-simplices.
Simplicial complexes are the backbone of computational topology because a finite complex is a finite data structure — a list of vertices and which subsets form simplices — so a computer can store it and run linear algebra over a field or over Z to get Betti numbers and torsion. The honest caveat is the cost of subdivision: a manifold may need very many simplices to triangulate faithfully, the number of high-dimensional faces can blow up, and not every topological manifold even admits a triangulation in high dimensions. Also do not confuse a simplicial complex with a simplicial set or a CW complex; the latter glue cells by arbitrary attaching maps and are far more flexible but no longer purely combinatorial in the elementary sense.
The boundary of a tetrahedron is a simplicial complex with 4 vertices, 6 edges, and 4 triangles (but no 3-simplex, since the inside is hollow). It is a triangulation of the 2-sphere S^2: its Euler characteristic is 4 - 6 + 4 = 2, exactly chi(S^2), and its homology is H_0 = Z, H_1 = 0, H_2 = Z.
Four triangles glued along shared edges build a hollow sphere; the alternating face count recovers the Euler characteristic.
The face-intersection rule is essential: a set of triangles that meet improperly (sharing part of an edge, or crossing) is not a simplicial complex, and its naive boundary computation will give wrong homology. When you only need the topology, the abstract version suffices; reach for the geometric realization only when you actually need a metric picture.