This is exercise 2 from Ch. 1 of "Computational Topology: An Introduction" by Edelsbrunner & Harer:
Consider a triangulation of a simple closed polygon in the plane, but one that may have interior vertices inside the polygon. A shelling is a total order of the triangles such that the union of the triangles in any initial sequence is homeomorphic to a closed disk. Prove that every such triangulation has a shelling.
It's not clear to me what is meant by "initial sequence." Does it mean any sequence of the triangles starting from the first one in the total ordering, or something else? This is not defined in the book.
I've drawn a polygon and a triangulation below (not using internal vertices) with a total ordering on the triangles. Is this total order a shelling?
So, later in the book they define a shelling of a $3$-ball as follows:
Let $K$ be a triangulation of a 3-ball, that is, a collection of tetrahedra sharing triangles, edges, and vertices whose union is homeomorphic to $\mathbb B^3$. No other, improper intersections between the tetrahedra are permitted. A shelling of $K$ is an ordering of the tetrahedra such that each prefix of the ordering defines a triangulation of $\mathbb B^3$, and $K$ is shellable if it has a shelling.
But again, I'm not clear on what is meant by a "prefix" of an ordering. I see that there is a thing called a prefix order, but I'm not sure how this is being invoked here exactly.