How are lone pairs of valence electrons related to the shapes of species in which all pairs of valence electrons are involved in bond information (eg. removing on atom from the vertex of the equilateral triangle in the central plane of a bipyramid)?
Molecular shapes are determined by a set of rules called VSEPR (valence shell electron pair repulsion). This states that valence shell electrons both bonding and non-bonding (lone pairs) will arrange themselves around the central atom such that the electron repulsion forces will be minimized. In other words, chemical bonds and lone pairs will arrange themselves so as to maximize the distance between them. The main difference here is that lone pairs do not count toward a molecule's named shape whereas bonds do. In the example you state, an atom surrounded by five chemical bonds (like PCl5) will take a shape called trigonal bipyramidal. If one of the Cl atoms is replaced by a lone pair, the resulting shape is called "seesaw". If another Cl is replaced, the resulting shape is t-shaped. For a visual representation of these shapes, see the second link below.