This will probably get pretty long because I'll be going into a lot of detail. If you just want the bare bones, just read the bolded text, it'll be the summarized version.
Right, so, our story begins in England in the 1800s, on the estate of the Joestar family. Here we meet Jonathan Joestar, only son of the wealthy George Joestar, a popular young lad who is in every way a gentleman, living happily with his dog Danny and his sort-of kind-of girlfriend (you know how they were back in those days), Erina.
Enter Dio Brando, and everything goes to shit.
Dio is the son of Dario Brando, a crooked, drunken innkeeper. When Jonathan was just a baby, the family was travelling through the countryside in the middle of a storm, and the cliff collapsed under their carriage, sending them tumbling down into the valley. Jonathan's mother died in the crash, and when Dario came along to steal from the bodies of the victims, he discovered that George and Jonathan had miraculously survived as well.
he also finds a mysterious stone mask among the Joestars' luggage. Make a note of this. It will be important.
George Joestar, badly injured, wakes in the middle of Dario robbing him blind and mistakes Dario's actions as a rescue attempt rather than thievery. He asks Dario's name so that he can properly reward him.
George gave him some money with which to start his own hotel. Unfortunately, Dario sucks at business, and loves to drink, and the business fails. On his deathbed, he gives Dio Mr. Joestar's address and tells him to go there after his death, and that the Joestars would take care of him.
George Joestar being the generous man he is, he does, and adopts Dio as his own son. This turns out to be a mistake.
Dio is, put simply, a complete asshole. All he wants is to discredit Jonathan, and then kill George, so that he will inherit the Joestar fortune. He starts by making Jonathan's life hell: he stuffs poor Danny in the furnace, he chases Erina off, he beats the hell out of Jonathan in the boxing ring and spreads false rumors about among the other bouys, turning him into a social outcast.
Eventually, even the somewhat dimwitted Jonathan works out what's happening and charges in to assault Dio physically. This is exactly what Dio wanted; he figures that if he crushes him now, it'll also crush his spirit or something. Unfortunately for him, he underestimated Jonathan Joestar: filled with righteous anger, he beats the crap out of Dio until dear old daddy shows up to break them up.
Jonathan gets the blame for the fight and is duly punished, but on his way out he notices something: that Stone Mask mentioned earlier was hanging on the wall, and was knocked down during the scuffle. When a bit of Dio's blood splashed on it (I TOLD you he kicked the crap out of him!), a set of coils burst out of the mask's edges, curling backwards into where the wearer's head would be. Curious!
Now we fastforward several years, to when the boys are graduating from high school. Dio makes every appearance of being good chums with Jonathan, to the point where even poor Jojo feels bad for not trusting him. He's almost certain that Dio was behind Danny's death, and he KNOWS that he mucked around with Erina...but he's being so gosh-darn nice!
It doesn't last. You see, their father George has fallen ill of late, and is now bedridden. Seems fairly normal for a man of his age, until Jonathan in thye process of researching the Stone mask (which comes from South America and was used in some sort of ancient human sacrifice rituals) stumbles upon some of the correspondance that Dario Brando sent to George Joestar right before his death, describing the symptoms that beseiged him...and surprise, surprise, they match the symptoms George is currently experiencing exactly. Jonathan reaches the obvious conclusion: Dio slowly poisoned his father to death, and is now doing the same to George Joestar.
He rushes out to confront Dio about this, and catches him in the middle of bringing their father his 'medicine' (which is actually poison). Another fistfighyt ensues, in which Jonathan beats Dio and takes the poison. He's going to go have it analyzed, to prove it. He then leaves after warning the servants to take precautions about what Jonathan ingests.
He heads to london, where he meets and befriends a street ruffian by the name of Speedwagon, and tracks down the man who sold Dio the poison.
Shortly after entering the city limits, he's set upon by a mugger. Deciding that this is the perfect opportunity to test out the mask before putting it to use, he slaps the mask onto the mugger's face and draws a bit of blood. Spikes go into the mugger's brain and kill him.
Problem is, he gets back up about thirty seconds later.
Dio miscalculated again! The mask doesn't kill people: it turns them into the undead!The mugger attacks Dio again, and his strength is not monstrous. He's a vampire! Dio would have died right then and there if the sun hadn't risen, burning the vampire away into ash.
Understandably shook up, Dio returns home only to discover that Jonathan has beat him there. He has the poison-seller in custody, and he's singing like a canary, and something like ten police officers show up to arrest him. A scuffle ensues, ending with Dio slapping the mask on himself and trying to stab Jonathan in the chest to activate it. George takes the blow for his son and that works just as well for Dio's purposes. Dio gets shot about a dozen times and falls out the window, but pops back in the window right on schedule and sets about slaughtering the local police force.
Jonathan and Dio fight, and Jojo is at a clear disadvantage, but some quick thinking and a bit of luck allows him to trap Dio in the lower portions of the mansion, which he has set on fire. Jonathan escapes with burns all voer his body, and Dio is dead.
Sadly, not quite.
The poison seller finds and rescues Dio's deformed body from the rubble. He's horribly burned and can barely move under his own power, but he heads into the german hinterlands and begins draining people of their life energy in order to regenerate his own badly damaged body.
Meanwhile, Jonathan is approached by a man named Will A. Zeppeli, who is familiar with the Stone Mask (had a bad experience as a kid) and who informs him that he believes that Dio is still alive. He trains Jonathan in a technique for battling the vampires, known as the Ripple (or Hamon). In a nutshell, you use a special breathing technique to generate ripples of positive energy, which you can transmit through your fists. It has a variety of uses (healing being the first one introduced), but most importantly, when fired into a Vampire these ripples emulate the energy of the sun, destroying them.
So long story short (LOL TOO LATE) Zeppeli and Jojo travel to a fictional village in Germany, fight a bunch of vampires and zombies, Zeppeli dies, Jonathan sends a ripple into Dio killing him. Our Heroes return home, and it's a happy ending! Jonathan marries Erina (they had a scene earlier, it's not completely out of nowhere), everyone lives happpily ever after.
Now hold on. This is where shit gets WEIRD.
Jojo and Erina's wedding party. On a boat in the middle of the ocean (they're headed for America!) Suddenly BAM, we've got a party crasher! It's Dio--or, more specifically, it's Dio's DISEMBODIED HEAD.
Yeah, that's right. To avoid being destroyed by Jojo's Ripple attack, he TORE OFF HIS OWN HEAD and threw it away. That asshole poison salesman saved him yet AGAIN.
Now he needs a body. And not jsut any body: he wants JONATHAN's body. Although he looked down on him before, he now sees Jonathan as his one true equal in the world. He thinks that stealing his body will make him stronger than ever.
He manages to get the drop on Jonathan (attacking by shooting fluid out of his eyes...no, really) and punctures a lung, preventing him from using the Ripple. He also brings a whole bunch of zombies to the party, who run around slaughtering the guests and crew.
With his dying breath, Jonathan manages to force one of the zombies into the engine in a way that causes it to overheat and explode. The ship sinks with Dio's head and Jonathan's body on board. Erina is the only survivor, riding out the blast in the coffin that Dio snuck abord in. She also saved the child of one of the guests (and is also pregnant with Jonathan's child).
And that's the end of part one!
TL;DR VERSION: Dio wanted to steal the Joestar fortune, but fucked it up and so turned himself into a vampire, was eventually defeated by his adoptive brother Jonathan Joestar, and then came back as a DISEMBODIED HEAD, sucker-punched Jonathan, and they both sank down to the bottom of the sea. Jonathan's wife, unborn child and one totally unrelated child survived.
Now, part two is about Jonathan's grandson, Joseph Joestar. He has his own adventure, and it's pretty awesome, there's cyborg Nazis and stuff, but none of it's important to Stone Ocean.
Part three picks up with Joseph's Grandson, Jotaro Kujo. Joseph Joestar makes a return appearance, and informs Jotaro that Dio has returned from his watery grave, now in full possession of Jonathan's body. He's also acquired a special ability, called a 'Stand', which I'll give more information on later. Because All other members of the Joestar line are also affected and given Stands, but Jotaro's mom can't handle the mental stress and it manifests as a fever that will kill her in fifty days if they don't kill Dio. So Joseph, Jotaro, and some other Stand Users depart Japan and head to Cairo, where Dio is gradually gaining strength. Along the way they fight various Stand users that Dio sends against them, until in the end they face off against Dio himself, and his Stand, The World. Better known as ZA WAAAARUDO.
The World allows Dio to stop time for several seconds. Dio can move, but nobody else can, and they can't even perceive what happens during the time stop. The 'few seconds' is subjective from Dio's perspective. This ability makes him damn near invincible, especially when combined with his vampiric strength and regeneration.
During the fight, Jotaro discovers that his own Stand, Star Platinum, also has the ability to stop time. It can't do it for as long as The World can, but it gives him the edge he needs to pummel the crap out of Dio, disabling him so that the sun can do the rest.
TL;DR VERSION: Dio returns, this time with Stand powers. Jonathan's Grandson Joseph, and his Great-Great-Grandson Jotaro, do battle with Dio again. Dio's Stand lets him stop time, but fortunately Jotaro's has the same ability (albeit lesser), allowing them to emerge victorious. Dio is now actually dead.
Alright, so, what is a Stand? A Stand is a concept introduced in part three, and used in every part thereafter. It's basically a manifestation of one's spirit. They often take on a humanoid appearance, but don't necessarily do so. One's Stand is invisible until it is called out by the user, at which point it becomes visible only to other Stand users. A Stand can also onloy be hurt by other Stands. Any damage done to a Stand reflects simultaneously onto the user as well, and vice-verse. For example, if you tear off a Stand's arm, the user's arm will also pop off.
Stands can be divided into a three different types:
1. Power-type stands. These have a very limited range, and have to actually touch you to hurt you. However, they have the most physical power. An example of this is Jotaro's Star Platinum, which hits very hard and very fast and very precisely, but only within a range of about two meters.
2. Remote Stands. These can travel away from the User for a long distance, and are controlled remotely by the user. However, The farther they travel, the weaker they get, and they aren't generally as powerful as Power-types. An example of this type is Kouichi's Echoes Act One from part four: it can travel up to thirty feet away from him and is frequently used for scouting.
3. Automatic Stands. These can go as far from the user as they want without having any diminishment in power or ability...but the User has no direct control over their actions once summoned. And example of this type is Sheer Heart Attack, one of Yoshikage Kira's abilities from part four. It's basically a miniature tank that perpetually travels towards the nearest heat source and then detonates, destroying whatever it was. It will continue to do this over and over until Kira recalls it.
You get a Stand by being pricked with one of a very few special Stone Arrows. If your spirit is strong enough to handle a Stand, you acquire one and the wound heals immediately. If it isn't...well, you catch a fever, break out in boils, and then die. By the time Stone Ocean rolls around, almost all of these arrows have been destroyed.
A FEW OTHER NOTES
- In the Jojo's Bizarre Adventure universe, what happens to one member of a family will affect all the other members, assuming that they are spiritually aware (ie Stand users). So when Dio acquired his stand, because he was using Jonathan Joestar's body, it rippled outwards through the rest of the Joestar bloodline.
- All Joestars have a birthmark in the shape of a star on their left shoulder. If they're Stand users, they can sense each other's general status and position through this.
- You remember Speedwagon from part one? Well, by the time part Two rolled around he was a wealthy oil baron, and from part three onwards the Speedwagon Foundation plays a large role in supporting the various Joestars.
- Rohan Kishibe is a manga artist from part four. This is only important to understand a cameo made near the end of Stone Ocean.
Stone Ocean scanlations can be found at Jojo Project. The first three volumes were done by another group, sadly, but the quality is on the rise from there.