Mines are cloaked, immobile structures. They deal damage to any ship moving over them and self-damage in the process. Both AI and Human players can use them. Properly used, they can be great defensive means.
Enemy ships coming too close to these invisible traps will incur heavy damage. The minefield is also damaged, and eventually destroyed, in the process.
The explosions from these minefields also affect other enemy ships that are close by. These are obviously useful for taking out groups of ships, but they are also useful for hitting ships that have Mine Avoidance -- those ships can still get caught in the blasts of area minefields that are triggered by another ship.
If the ship triggering the minefield survives, it will also be paralyzed for 10 seconds and suffer engine damage. The burst of paralysis and engine damage (but not the hull damage) also affects other ships that are close by. These are particularly useful for stalling out large incoming vessels that don't have mine avoidance or paralysis-immunity.
Some ships have the "Mines" immunity. They are essentially raid-like units. They can pass through minefields without triggering them.
Human players can build regular minefields from game start, but need to invest Knowledge in order to unlock area and widow minefields. They have no prerequisite and unlocking one doesn't require to unlock the other.
Minefields have a galactic cap, which mean only a small amount of planets can be defended that way.
Minefields are better used when placed on enemy path. Often, that path is between a wormhole and an Orbital Command Station. The line-place option is useful to automatically place mines in such path.
Cheap units can also be sacrificed and sent first through minefields in order to destroy them, tanking damages for more valuable units.
When allowed to use them, the AI often places minefields as packs around wormholes and Orbital Command Stations. AI minefields are automatically destroyed when the planet is captured by Human players.
All minefields turn into remains when destroyed (by gunfire or self-damage). AI players don't rebuild them, but Human players might use Remains Rebuilder drones in order to rearm them.