The stability of a graph is the maximum size of a stable set, i.e. a set of vertices which are not pairwise linked by an edge. In particular, stability 2 means that G is the complement of a triangle-free graph. A matching is a set of disjoint edges. A matching is complete if the contraction of its edges gives a complete graph.
The fact that this conjecture is called the seagull conjecture is left to the reader as an easy exercise.