Mines are small devices filling a circle around a point in space. Any enemy starship passing through a minefield has a chance of hitting a mine. A mine hit can do a significant amount of damage to a starship (destroying smaller starships) and could slow it down.
Any starship may set the Mine Sweep mission. During the Host run, each starship with this mission set will scan local space for any minefields whose edges are within 200 Light-Years of the starship and issue a report of its findings. This mission is used both to sweep (reduce/destroy minefields of unfriendly Commanders) and to scoop (convert your minefields back to torpedoes), depending on the Friendly Code of the starship.
Details of the process and effects are described in the minefields page.
See also: Out of Fuel.