Luck Points in FS3 are a way to give your character an edge from luck or fate.
You can spend Luck Points in various ways to help your ability rolls.
Only one Luck Point can apply to a given task, so you can’t give yourself a bonus and then also re-roll the result.
You can spend a Luck Point for special effects. See Combat.
You can also spend Luck Points to affect the outcome of plots. For example: you might spend a point to have the prosecutor lose key evidence in a trial, or to get a lucky break finding a missing witness. This is entirely at the storyteller’s discretion, and the number of points required will vary based on how much you’re pushing your luck.
How you gain luck will vary from game to game. See FS3 Core for an example.