Nanite Repair Paste is an NPC-seeded item type added to the game in the Trinity expansion used for in-flight repairs to modules damaged by overloading, a function of the Thermodynamics skill. After the Tyrranis expansion, nanite repair paste became a produceable commodity through planetary interaction. Nanite Repair Paste does not require any skills to use, but its use is affected by two: Nanite Operation and Nanite Interfacing, which modify consumption amount and repair speed, respectively.
If you already use Thermodynamics at all, Nanite Repair Paste will be your new best friend. If you don't use Thermodynamics, but can, you should give it a try now that module repairs can be effected anywhere, even while cloaked.
For more detailed information on what overloading is and what to use it on, see the main Overheating article.
To use Nanite Repair Paste in your cargohold to repair a damaged module, right-click the module on your HUD and select Repair. It will start flashing brightly, and when the repairs are finished (or canceled), it will consume paste from your cargohold. If you don't have enough paste to affect repairs of the selected module(s), it will tell you as much and will not begin repairs.
Nanite Repair Paste is a produceable commodity through planetary interaction. It is a level 3 commodity and requires the purchase of the nanite repair paste BPO to produce.
Again, the quantity of nanite paste consumed to repair an item is dependent on its base cost. You can easily find the base cost of a module by looking it up in the Item Browser subsection of Evemon's skill plans.