Diagram of electrostatic precipitator
Looks like two positively charged plates with negatively charged rods spaced apart in between. Air flows in, corona discharge strips the air molecules of electrons which ionizes them, and the free electrons bounce around and hit the dust particles and negatively charge them. The now negatively charged dust particles are attracted to the positive plates out of the air.
Both plates are positive because it's a lot easier to charge the dust negatively than it is to charge it postively. You want to maximize surface area to attract the dust and the maximum surface area is from the plates. So the plates are positively charged. The negative electrodes need to be narrow to promote coronal discharge (by concentrating the electric field like a lightning rod). Since the negative eletrodes can't attracting dust anyways, the narrower the better.
The coronal discharge produces free-electrons by stripping the air of electrons thereby ionizing it, and the free electrons stick to the dust particles thereby negatively charging them.