Which state of matter is characterized by ionized particles, no definite shape or volume, and good electrical conductivity?

Answer is: plasma.

There are five fundamental states of matter (gas, liquid, solid, plasma and Bose–Einstein condensate).  

In solid, molecules are closely packed, stiff and do not changes of shape or volume. Solid object (for example iron) does not take on the shape of its container.  

Liquids have definite volume, but no fixed shape.  

Gases (for example nitrogen and neon) not have definite volume and fixed shape, it depends on its container.  

Plasma is a an ionised gas with highly electrical conductivity.

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