Quests are (upcoming) assignments you can obtain to gain cash for ship modules or other rewards. Unlike other games, LSF lacks a conventional mission posting. Instead, you will acquire quests by keeping an eye to the ground, watching news tickers, reading direct messages from pilots you encounter and so on, then traveling to the target area.
Quests aside from the tutorial have been disabled since version v1.09 due to incompatibility. They will be reimplemented at a later date.