Orbital Refinery provide bonuses to Neutronium and Lommite production.
Orbital refineries work by sending out ships to adjacent planets and increasing production at the extractors there.
Each refinery servicing an extractor at one of your planets earns it an extra 0.06 metal or crystal/sec. There can be a maximum of 3 refineries servicing one extractor. Thus, saturating your empire with refineries increases your income by 45%. It is unknown whether this is affected by allegiance; if it is not, it would be more.
Each refinery servicing one of your extractors at uncolonizable planets earns it an extra 0.08 metal or crystal per second. There can be a maximum of 4 refineries servicing one extractor.
It can be a mini-game unto itself to place refineries most efficiently. Here is an example:
C | A--B--F | E
The optimal strategy here is to place three refineries at A and three refineries at B. This will cause each planet to be serviced by exactly three refineries. This is, however, a very simple layout, and there will often be no perfect way to place them.
A somewhat good way to place them without worrying about optimality is to just put one per planet. That's simpler, but doesn't work as well.
Another way to do it is to look for a planets that have several planets near them and place three there.