A fortnight is basically like a chapter in story mode, except that it's focused on one character or one specific part of a storyline. Some characters exist in the story mode like Alvida/Buggy/Smoker, but some are exclusive to their respective FN and can only be farmed during this time. Also, each FN has skillbooks, which you use to lower a unit's special cooldown.
So like, take Alvida's FN. It's like 5-7 battles, with Alvida being the last boss. You fight through stages of fodder guys, and the sub-boss, who is Coby in this FN. They can all either drop their poster, beli, or nothing. Some characters can also drop skill books, and different difficulties can yield different skill books. For example, you can farm Zoro books in Alvida's FN, but if you want Ashura Zoro skillbooks, you need to run the hardest one to get have the chance at that. There are also rare encounters that don't show up every time, but when they do, they'll drop a poster. In the case of Alvida's FN, it's Gaimon or Master of the Near Sea.
The Alvida FN is a pretty simple example, because it was one of the first ones and there weren't too many units in the story by that point, nor in the game.
Jump ahead to the Bride Nami FN that's going on now, and you've got Bride Nami as a boss, Lola as a sub-boss, and the Risky Brothers as the rare encounter. But, you can also have a final stage where Nami appears with Lola, or with Absalom, or with Absalom and Sanji. So it gets more varied as it goes on.
The main one you'll want to look out for is the Captain Kuro FN, because that's where you can farm GP Usopp books, specifically on the 15 stamina tier. Having his special maxed is pretty key for a lot of content, and can save your skin fairly often.