Lost Sector
Lost Sector is an adventure puzzle mode where the main goal is to kill all enemies and find all treasures hidden on the map.
When you unlock this mode - after clearing Stage 5-3, only the first Sector will be available and you have to clear it and progress further in the Campaign to unlock the other Sectors.
Rewards
The rewards for each Sector are fixed and to get 100% score, you need to gather them all. So it's advised to try to clear the Sectors as much as you can the moment you unlock them.