Repairing
You are able to repair some things that take damage. The most common ones are floors/walls you build, and your ships. When your ship takes damage, you will see the tiles turn from their normal verion, to a damaged version.
Right click on the damaged tile, and select 'Repair'. You will start repairing the tile. Things that aren't ships generally won't have a damaged/undamaged version of a tile.
Repairing requires material. Repairing will always require the material that the object was made of. If the object was made of multiple things. In the example of the Fighter ship above, you would need Refine Territe in your inventory in order to repair it.
In the case of the Lancer which is made from 40 Exodium and 8 Power Cells, the game uses what it is made mostly of (Exodium) for the repairing process.