When drawing Lewis structures, sometimes you will find that there are many ways to place double bonds and lone pairs about a given framework of atoms. How does we decide whether one or another placement is correct? The answer, as it turns out, is neither and both. The actual arrangement of electrons in a given molecule is a weighted average of all the valid Lewis structures that can be drawn for that given atomic connectivity. The "real" molecule, the one that actually exists in the world, is said to be a resonance hybrid of all its contributing Lewis structures. Each Lewis structure that contributes to the resonance hybrid is a resonance structure.
The classical example of resonance is benzene, C6H6. Two good Lewis structures for benzene exist that differ only in their placement of double bonds. If either structure were correct, benzene would consist of alternating long single bonds and short double bonds. However, it has been determined experimentally that all six bonds on the ring are identical. The natural interpretation is that the three double bonds are distributed evenly around the ring, so that each bond has a bond order of one and a half.
A double headed arrow is placed between resonance structures to denote them as such. In addition, sometimes we place all the resonance contributors within brackets for clarity.
It's important to remember that although the molecule described by resonance has characteristics of all its resonance contributors, it is fully neither one. For instance, the color gray might be described as being a resonance hybrid of white and black. And although gray takes on characteristics of both black and white, it would be incorrect to describe gray as being black or white.
Sometimes double-headed arrows are used to denote how one resonance structure can be derived from another via the flow of electrons. This curved-arrow formalism is a very useful bookkeeping tool that allows us to keep track of the movement of pairs of electrons during reactions. The arrows are drawn from the source of the electron motion, which can be a bonded electron pair or a lone pair, to the destination of the electrons, typically an atom or a place between two atoms. The figure below illustrates correct and incorrect usages of the curved-arrow formalism. We will see that it is a very useful tool for describing reaction mechanisms, the step-by-step processes by which reactions occur. Such exercises are affectionately referred to as "arrow pushing".
Resonance is such an important concept to master early on in your organic chemistry education that it's worthwhile to clear up two potential misunderstandings. Resonance and equilibrium, and resonance and isomerism are often confused.