Mining drones are drones that you can deploy at an asteroid field which will automatically mine a type of ore each hour and drop it at an orbiting refinery, (standard drones will mine standard ore, advanced drones will mine other things and require a Advanced Merlin Refinery).
The number of Mining drones you can have active at one time is dictated by your number of free drone control points (you have 10 by default or 15 with the Drone Point expander upgrade from the CE Store).
Combat drones you need to launch yourself during combat, they will attack the enemy ship and do damage each turn, though the enemy can also shoot at and destroy them. Combat drones can be used irrespective of drone control points.