Edge collapse
The atomic operation of mesh simplification: an edge is contracted to a single vertex and two triangles vanish.
Collapsing the edge between two vertices merges them into one point; the two triangles that shared the edge degenerate and are removed. The operation is reversible — a vertex split recreates the edge — which is exactly what a progressive mesh exploits.
Which edge to sacrifice, and where the merged vertex lands, is decided by an error metric, most often quadric error metrics. The pitfalls are topological: a collapse must not fold the surface back on itself or create non-manifold edges, so most implementations reject collapses that would flip a triangle's normal.