At atmospheric pressure, air and other gases are poor conductors (Insulators) of electricity. Because, they don't have any free electrons to carry current. But, once the free electrons are produced in gas by ionization (they become plasmas), electric discharge occurs.
This could be done in many ways such as by applying a large potential difference across a gas column at very low pressure, or by allowing high frequency EM-waves such as the X-Rays through the gas. This question is not proper to ask. Indeed, gases become plasmas once they're ionized!
All dielectrics have a certain value of breakdown potential. In most of them like air, ceramics, etc. (including semiconductors like Silicon), this potential provides sufficient energy to ionize a few atoms. The free electrons formed as a result of this ionization are energized enough to bump the neighborhood atoms, break some covalent bonds and produce more free electrons.
These free electrons are then accelerated by the applied electric field and they collide and ionize the other atoms to produce more free electrons (multiplication by collision). Now, there are large number of free electrons available for current flow. Thus, an electric arc is produced!
However, this isn't applicable for all gases. As @mikuszefski has mentioned, there are a few other cases such as the noble gases which are mono-atomic (and are used in most of the discharge tubes), whose breakdown potentials are only meant to ionize them!
This video illustrates how a candle flame conducts a high voltage (10,000 volts). Although the ion density is small in a candle flame, they are sufficiently present to conduct electricity.
Best Answer
This is due to the principle of dielectric breakdown. During thunderstorms, the air between the cloud and the ground acts like a capacitor. When the electric field is high enough, the air partially ionizes, at which point there are free electrons to carry current and the air becomes, essentially, conductive.