Air is made of molecules, and the molecules are made of protons, neutrons and electrons. Whenever a spark appears, those same protons, neutrons, and electrons are still there. There's a difference of course. In air, the protons and electrons are bound together, so they cannot flow separately. Within the spark the electrons and protons are no longer stuck together, and electrons can flow through the cloud of protons, and vice versa.
Sparks are simply regions in the air which has undergone a change in phase.
Sparks however are conductive, while air is not. To answer this we need to look at the microscopic details of spark creation. When you apply a large voltage between the metal parts of a spark plug, you create an electrostatic field in the space between the metal parts. However, the net voltage placed across the metal electrodes is not the only thing that determines the strength of the e-field. The e-field will be strong near small, sharp electrodes, and it will be weaker near the surfaces of large flat ones.