Resonance is to be distinguished from isomerism. Isomers are molecules with the same chemical formula but are distinct chemical species with different arrangements of atomic nuclei in space.  Resonance contributors of a molecule, on the other hand, can only differ in the way electrons are formally assigned to atoms in the Lewis structure depictions of the molecule.  Specifically, when a molecular structure is said to be represented by a resonance hybrid, it does not mean that electrons of the molecule are "resonating" or shifting back and forth between several sets of positions, each one represented by a Lewis structure.  Rather, it means that the set of contributing structures represents an intermediate structure (a weighted average of the contributors), with a single, well-defined geometry and distribution of electrons.  It is incorrect to regard resonance hybrids as rapidly interconverting isomers, even though the term "resonance" might evoke such an image.  (As described below, the term "resonance" originated as a classical physics analogy for a quantum mechanical phenomenon, so it should not be construed too literally.) Symbolically, the double headed arrow  is used to indicate that A and B are contributing forms of a single chemical species (as opposed to an equilibrium arrow, e.g., ; see below for details on usage).
A non-chemical analogy is illustrative: one can describe the characteristics of a real animal, the narwhal, in terms of the characteristics of two mythical creatures: the unicorn, a creature with a single horn on its head, and the leviathan, a large, whale-like creature.  The narwhal is not a creature that goes back and forth between being a unicorn and being a leviathan, nor do the unicorn and leviathan have any physical existence outside the collective human imagination.  Nevertheless, describing the narwhal in terms of these imaginary creatures provides a reasonably good description of its physical characteristics.
Due to confusion with the physical meaning of the word resonance, as no entities actually physically "resonate", it has been suggested that the term resonance be abandoned in favor of delocalization and resonance energy abandoned in favor of delocalization energy. A resonance structure becomes a contributing structure and the resonance hybrid becomes the hybrid structure. The double headed arrows would be replaced by commas to illustrate a set of structures, as arrows of any type may suggest that a chemical change is taking place.