KoF Sagas. Been a while since I played the series, so totally open to corrections. 
I unfortunately missed out on KoF after 2003 so, I only have vague ideas on what happened.
Rugal Saga:
KoF 94
Kof 95
This is the intro. A mysterious benefactor hosts the latest King of Fighters and invites fighters from all over the world to compete. This is in a different continuity, but it unites Fatal Fury, Art of Fighting, revamps Athena and the Ikari warriors.
The Hero team is made of Benimaru, Goro and our resident hero Kyo. Kyo is from the fabled Kusanagi bloodline that helped to banish the powerful creature (god?) Orochi. They hold a seal within their bloodline that allows them to create fire , not ki-infused fire, but totes for real scorchy scorchy fire.
Their rivals were the Yagami clan which partnered with Orochi for power in exchange for a curse. They received powerful purple flames. Iori’s Maiden Masher is referencing the 8 maidens they had to sacrifice to Orochi to appease it.
Anyways, Rugal Bernstein is a legendary arms dealer and fighter. He has a reputation for dipping defeated enemies in molten steel/bronze WHILE THEY’RE STILL ALIVE to cultivate a collection. Rugal is a natural talent and has an ability to copy other fighter’s styles with little effort. He has copied and utilized Krauser’s Kaiser Wave and Geese Howard’s reppuken.
Before the events of KoF 94 he made a discovery of a powerful source of energy and went to investigate, finding Gonitz. Goenitz literally yanked out his eye, but was surprised how resilient Rugal was. Rugal wanted more power and was given some of the orochi power despite warnings that it would destroy him.
The finalists of the 94 tournament were taken to the BlackNoah, Rugal’s personal airship and were engaged in an intense battle which culminated with Rugal detonating the whole thing since he was a sore loser.
More invites went out the next year. It turned out that Rugal was quite alive and still burning with revenge. Kyo’s father was captured and brainwashed by Rugal’s secretary in KoF 95. I think it was Vice, but anyways his two secretaries - Vice and Mature were in reality Orochi spies working for Goenitz.
A new team arrive headed by Iori Yagami, Billy Kane, and Eiji. Iori is Kyo’s rival and slowly, the main plot points of the series begin to emerge. Canonically the hero team makes it to Rugal once more, amazed that he survived an exploding Airship. The only evidence of his injury is a bionic hand that replaced the one he lost in the blast. He mocks and toys with the team by first having Kyo’s brainwashed father Saisyu, fight the team. Kyo eventually wins much to Rugal’s amusement.
He then reveals he has the power of Orochi and taps into it’s ancient wellsprings. His hair becomes white, his skin grey and his overall demeanor demonic. He’s ludicrously powerful but eventually gets beaten. Trying to utilize all of the Orochi power Rugal begins to break apart. Before his death, Iori pops in and calls him a fool.
Rugal was able to utilize the power of Orochi because he was a powerful fighter, but since he did not possess the bloodline of the Yagami’s or be born as a lord of Orochi, he would be consumed by it’s power.
Orochi Saga:
96
97
Here’s were things really take off. Goenitz is the host of the 96 tournament. We see more new faces and the return of some old enemies. Rugal is gone and Iori is joined with Vice and Mature. To make a long story short. Goenitz reveals more about the orochi bloodline and upon his defeat Iori and newcomer for the Ikari team, Leona are tainted. Both being tied to Orochi have had their Riot of Blood unlocked. It wont fully come into play for Leona, but Iori is driven mad and kills Vice and Mature. The Riot of Blood affects those connected to Orochi and drives them into a state of incredible power but feral madness. The horrifying thing is that Ryuji Yamazaki is a Lord of Orochi but doesn’t seem at all different because his default state is so blood thirsty, insane and sadistic.
In 97 the New Face team, a young group of musical rivals to Iori’s band show up. Something seems off about them, but then it turns out that they’re lords of Orochi and are tasked with reviving him once more. The tournament erupts into absolute chaos. Leona, to her horror completely loses control of herself and becomes feral in the Riot of Blood stage as does Iori. It’s implied that she kills quite a bit of innocent bystanders.
Kyo’s girlfriend, Yuki is saved from being sacrificed. It turns out she’s a descendant of the last sacrificed maiden.
-98 First Dream Match, a break for the new saga
Nests Saga:
KoF 99
KoF 2000
KoF 2001
-KoF 2002 Dream Match, saga break
KoF 2003
KoF XI
KoF XIII
will post more later