Ozone is triatomic oxygen, O3. It has two resonance structures as shown in the drawing below. The three oxygen atoms have a total of 18 valance electrons. If both outer electrons were single-bonded to the central oxygen there wouldn't be enough electrons to complete the octet for all three oxygens, so a double bond is required for a valid Lewis dot structure. The double bond can be between oxygen 1 and oxygen 2, or between oxygen 2 and oxygen 3. Each oxygen has either 1, 2 or 3 lone pairs of electrons to complete the octet, as shown in the image. The two different possibilities are called resonance structures. The actual bonding in O3 is a hybrid of these two structures. A double bond consists of a sigma bond, which has its electron density mostly between the two bonded atoms, and a pi bond, which is delocalized (has its electron density spread around more.) The molecule can be though of as having two single bonds and a 3rd bond that's somewhat shared by all three atoms.
Resonance structures got their name in an attempt by early chemists to explain how two identical atoms with equal electronegativity could have different numbers of shared electrons and lone pairs in a Lewis dot structure. It was hypothesized that the atom rapidly flipped between the different structures, vibrating or resonating in the process. Resonance structures are now viewed not as two or more forms in which the atom can exist, but as the best way to describe bonding that's a hybrid of the possible structures.