There's no such thing as a quick summary for these games, so let me get down to business. Warning, it's long, but it's all valuable. Read it two or three times if it doesn't all click right away, and feel free to ask me some questions or to refer to a wiki to fill in the gaps wherever they may be. You basically just asked for the whole story though, and there's no way to keep it short and sweet. It's honestly rather confusing at first, but you get used to it after a bit of reading.
You played Birth By Sleep, so that saves a little bit of explaining. Ventus's heart was totally broken after the events of Birth By Sleep. In order to repair it, he placed his heart into a very young Sora who's heart called out in response to Ventus's heart seeking a place to rest and fix itself. Ventus's body is housed within Castle Oblivion, a castle explored in Chain of Memories, which I'll touch on later. The Organization is comprised of just about all of the special people you meet in that game. It's the first and only time that the developers reveal the true names of these individuals, or have them shown in their pre-Nobody form.
[Heartless are what are created when the darkness in a person's heart overwhelms them, resulting in the loss of their heart]
[Nobodies are created at the same time as a heartless. The Nobody is the left over physical form. They have no emotions (any emotion shown is considered fake). Nobodies don't seem to spawn in the same place as the heartless. Rather, they seem to end up elsewhere.]
In Kingdom Hearts, Sora starts on Destiny Islands with Riku and Kairi. The three are eventually separated when Destiny Islands is taken by the heartless. Riku accepts the darkness, while Sora chooses to fight it, acquiring the keyblade shortly before the islands fall into oblivion. Awaking in Traverse Town, he eventually meets Donald and Goofy and journeys to each world trying to stop the heartless. His efforts prove successful, and he runs into Riku several times throughout his journey. It's found that Riku has descended deeply into the darkness, helping to collect the seven princesses of light. These princesses (Disney princesses + Kairi) are supposed to be the key to unlocking Kingdom Hearts. Eventually, Sora fights Riku and uses the dark keyblade in order to release his heart (which, much like Ven's, seems to have trapped itself in Sora in order to stay safe). Remember, Sora has Ven's heart in him, so this also must be released (a detail learned long after the release of KH 1). Sora becomes a heartless with the loss of his heart, and presumably Roxas is simultaneously created elsewhere. Eventually, with the aid of everyone, Sora regains his heart and turns back to normal, sans Roxas and Ven's heart. After this all occurs, you fight who, at the time, you believe to be Ansem, but who is really Xehanort's heartless. At the end of the conflict, you meet King Mickey for the first time in order to seal Kingdom Hearts. Riku ends up in the realm of darkness with Mickey, while Sora locks the door from the other side. The game concludes with Sora, Donald, and Goofy hunting for the King and Riku, who went to hunt for the exit from the realm of darkness.
This leads into Chain of Memories, which serves to give a loose introduction to Organization XIII, but more exists to simply bridge KH1 and KH2. You kill some members of Organization XIII as both Riku and Sora (two separate stories) without really knowing who they are. Kairi's Nobody, Namine, is manipulating Sora's memories throughout the entire game, as she is forced to by the Organization. As for Namine, she was likely created whenever Kairi originally lost her heart. Since her heart fled her body, no heartless was created (there was no darkness to produce one within Kairi), but a Nobody was still formed. Sora's story ends with him being placed in a kind of cryogenic sleep of sorts for Namine to fix his memories. Riku's story ends with him meeting DiZ (Ansem the Wise - the true Ansem), and learning to wield his darkness.
*NOTE: The Organization was in Castle Oblivion for two reasons. Most members were there simply to be eliminated by Axel. They were seen as a liability, and Xemnas wanted them gone. The second reason was that they were hunting for the room where Aqua stashed Ven's body before manipulating her, Terra's and Ven's old home to fit the current needs.
*NOTE 2: The Organization is made up of 13 members. Each member is a Nobody that was part of Xehanort's life before his experiments begin, resulting in all of the chaos that has occurred. You meet a number of these members before they become Nobodies in Birth By Sleep. Roxas was the only member of the Organization that was not linked to Xemnas's life when he was still Xehanort. Roxas was accepted only because he could wield the keyblade. The keyblade gathers hearts, and the Organization were working to recreate Kingdom Hearts so that they could have hearts of their own, to become people once more. Sora and Riku ultimately stop Xemnas and the rest of the Organization.
*NOTE 3: If it isn't clear, Xehanort was the apprentice to Ansem (Ansem the Wise). After everything got screwed up, Xehanort lost his heart. His heartless was formed, who parades around as Ansem in Kingdom Hearts 1 and acts as the final boss. His Nobody, Xemnas, was also formed. Xemnas goes on to create the Organization, and is the final boss in Kingdom Hearts 2. Something fun to note on this note: when a person's heartless and nobody are both destroyed, apparently the remnants rejoin. Now, remember, Xehanort was originally Terra-Xehanort, after Master Xehanort forced his heart into the body of Terra. If he were to reform with his memory regained, the effects could be devastating.
This leads into Kingdom Hearts 2, where you start off playing as Roxas. Roxas is an oddball nobody, as not only can he wield one keyblade, but he can actually wield two. Not only that, but he does actually have emotions to some degree. The first keyblade can be attributed to the fact that he's a keyblade wielders Nobody. The second keyblade, as well as his appearance, is due to him actually containing Ven's fractured heart, which is his acting heart, but a bit busted up still. Roxas is kept in a false Twilight Town by Ansem the Wise after being taken by Riku in The World That Never Was, home of the Organization. Ansem soon leads Roxas back to Sora's body, which has recently had all of its memories fixed and restored, and Sora becomes entirely whole as he once was before the events at the end of Kingdom Hearts 1 transpired. As well, Namine eventually rejoins with Kairi, though I forget exactly when that event occurs. Riku, at this point, is parading around in the form of Ansem's heartless after almost losing himself entirely to the darkness in order to do what he felt he needed to. His memories don't come back for awhile, but when they do, he aids Sora in secret and eventually is turned back to normal thanks to Sora at the end of the game. The game itself is spent fighting the Organization, Nobodies as a whole, and the heartless on the side. The game reaches its final climax in The World That Never Was as the rest of the Organization is defeated, and Sora and Riku kill Xemnas. The final cinematic has Sora, Riku, and Kairi back on Destiny Islands when a bottled message reaches them. This is presumably the letter offering the lead in to the 3DS entry, informing Sora and Riku of the Mark of Mastery exam.
*NOTE 4: Don't bother asking why everybody and their brother can seem to use a keyblade after awhile. Mickey, you just have to accept can use one. Riku inherited Terra's legacy (maybe that explains why Riku is so susceptible to the darkness from the get go). Sora was chosen by the keyblade, but he also had Ven's broken heart, which may have explained that one. Kairi, we just assume it's because she's pure of heart, or maybe because she has such strong ties to Sora and Riku. Thanks to beta4attack: Kairi touched Aqua's keyblade when she was very young back in Radiant Garden, initiating the the inheritance of power.
