I will try to put it very short there.
Amakusa is executed by goverment forces for christian rebellion. He abandons christian god and mades a pact with dark god Ambrosia
Amakusa resurrects taking the body of Shinzo (hanzo’s son), and sends disasters, planning to eventually make the path to our world for Ambrosia. The magic gem he uses was stolen from Tam Tam’s village btw. Characters travel the world to stop him
SS Haohmaru Jigokuhen
Mizuki (posessing body of shinto priestess Bizuki) again sends demons and disasters to the world and wants to capture 4 sacred souls (Haohmaru Jubei Sieger Kyoshiro) to achieve power needed to led Ambrosia in human world. Characters travel the world and in the end must get in Makai (demon world) via Mountain of Fear and battle Mizuki to prevent her reaching human world.
I will note that Mountain of Fear (Osore Zan) at edge of japan right before ainu lands, you can say it stands between main japan and ezo, and is believed to have passage to Makai via its caves.
SS Zankuro Musouken
Zankuro is a giant warrior who is obsessed with perfection of his style to the point of maddness, he slaughtered everyone but then realised error in his ways and would attack only those who challenge him. Characters go to meet Zankuro for various reasons. (this is before SS Haohmaru Jigokuhen)
Amakusa’s soul split in 2 half, good and evil (blue-skinned boss one). Evil Amakusa summons his castle in Shimabara where he was executed; as well as summons Zankuro back from the dead, and sustains both with power of Hazuki Kazama. He plans to dominate the world, and other characters including good Amakusa go to stop him (after SS Zankuro Musouken but before SS Haohmaru Jigokuhen)
Mysterious demon wizard, Yuga (in next game “he” is revealed to be remotely controlled body and actual Yuga was female sorceress from Makai); he controls big amount of people like puppets (hence called puppetmaster) and his target is merge of Yin and Yang forces of universe (notably merging Human World with Makai), for that he needs to be reborn as new dark god of Makai. Since he uses male-female body merging technique/sacrifice to create new body, for dark god body he needs Woman of Darkness Half and Man of Light Half = Shiki & Haohmaru, and he captures Shiki succesfully
(Note: its heavily hinted that Yuga is related to Ambrosia)
Yuga’s big second comeback and at this time she (appears herself) and almost succeeds, with both Shiki and Haohmaru captured and main team of “good” guys unsuccesful/defeated.
Instead, Yuga gets defeated by her dreaded enemy, Makai swordsman, Asura; he was sealed but escaped when Yuga was defeated in SS64.
(Note: All the setting is based around Mountain of Fear again)
"New Generation" game a-la Garou or SF3, with mainly new storyline, and with subplot storyline arc (in which Yuga plays a big role since her spirit survived in Mikoto) to end SSAZ plot. Former leader of Iga Ninjas and now wandering bodhisattva (enlighted) cames to conclusion that there is need in radical change in Japan, which lost its martial spirit after years of peace. He plans to overthrow government and create a martial nation for the Bushin (martial arts god), nation where strong survive. For this he works through three mighty warriors called Three Blades of Domination, and also one of Tokugawa branches actually supports him. The center of all action is Riten-Kyo, a small prison island near Edo from where this new “wave” meant to be spread, as Three Blades of Domination start to act here; and various warriors gather on the island to encounter them.
SS Zero (before SS1)
There is famine in Japan, and since Tokugawa government does nothing, protector of Hinowa province, Gaoh decides to rebel and overthrow the government, and also he start recruitment of strong warriors. Though Gaoh is a hero he also appears to be posessed by ancient demon, called Kuraki-Sumeragi (Dark Emperor), who only seeks to start wars to get people die. Well a number of characters go to the Hinowa for various reasons
SS Tenkaichi Kenkakuden (uncanonical entry after SS Zero, a dream-match)
Yoshitora who is apparently a shogun (already), announces an Imperial Tournament (Gozen) for warriors all over the world, with big award. Matches are set during the various countries festivals (each arena in game = some festival). Apparently there is a rule to dont kill opponent (for Gozen’s it in fact optional but since the game doesnt have the gore…).
In the end though it seems tournament faces some disaster (this is during you battle one of previous bosses) and you end up in Makai, forced to battle Gaoh who was reborn as the demon lord (MaGaoh). (Well if Gaoh is based from Nobunaga then MaGaoh is based off resurrected Nobunaga from Onimusha. Simple). As I can judge, MaGaoh wants to conquer Makai, and Kuraki-Sumeragi is still with him as well.
SS Sen. (it seems it is between SS64 and SSAZ)
It begins from story of how king and queen of Lesphia (kingdom in Europe) were killed by assassins and one of guards escaped to Japan together with their only daughter, who became in the end foster daughter to lord of Amori clan. After some years when she grown, mysterious foreighn ship was spotted to constantly be near Japan, country became flooded with illegal foreighn warriors who waited their time, and samsho characters started to suffer from assassin attacks. Apparently main boss is one responsible for many murders/assassinations and he commands that ship. Characters go to investigate