the game starts with you playing as Desmond Miles, a New York bartender. You've been kidnapped by a pharmaceutical company called Abstergo, which is run by some barmy old codger called Warren Vidic (who just so happens to be a complete smug prick who thinks his shit don't stink). Vidic tells you that Abstergo is a famous company that has developed pretty much every invention ever. As you play the game, you find out that Abstergo is a modern day front for the Knights Templar, a faction who strive for world peace by taking away free will (they're the bad guys). You also learn that they think you're a member of the Assassin Brotherhood (you used to be, but not anymore), a faction that have been rivalling the Templars for centuries to try and stop their shenanigans (eeeevil shenanigans).
the reason they kidnapped you, is to place you in a machine called the Animus (which is actually Latin for soul). What the Animus does is allow you to relive the memories of your ancestors by accessing the genetic memory stored inside your DNA. They want to do this because your ancestor, an assassin from the 12th century called Altair Ibn La-Ahad, came into possession of an artifact called The Apple, which is one of many Pieces Of Eden. So without so much as a how-do-you-do, they (Warren and his assistant Lucy Stillman), place you in the machine.
when the Animus starts up, you take over as Altair at Solomon's Temple, looking for the Apple, but arrive just in time to see Robert de Sable, the Templar Grandmaster, get to it first. Because he's such a massive douche who stole your prize, Altair tries to stab him in the face, but fails. Shit hits the fan, and your buddies get caught behind as you're forced to flee the collapsing temple.
you return to the Assassin's castle stronghold at Masyaf (real place, used to actually be the stronghold of the Hashashin, the people where the word 'assassin' comes from). Your buddy who was left behind, rocks up minus a functioning arm with the Apple, with a Templar force right behind him. You do some sneaky bastard stuff, and sneak around them and overpower them.
after they've fucked off, your boss, another barmy old codger named Al Mualim, thinks you dun goofed, so he demotes you by stabbing you in the neck. Okay... aaanyway, somehow you survive, and are told you're going to have to earn you way back up to the top rank.
you are the provided with names of people who require murdering because they're being complete and utter bastards. You ride to three different cities; Jerusalem, Damascus and Acre (the game's set in the Holy Land during the Crusades. 1191 to be exact), and have a particualr target at each one, whether it be a weapons dealer, a lord, a military commander, or just some fat cunt with a squeaky voice. You gather info on your targets, and assassinate them one by one over the duration of the game. Every kill restores a level of rank (also, each target is based on an actual historical figure that died in 1191). As you notch up a bodycount, you notice that each of them, in their dying moments, all make a confession (this happens with every main target in each game of the series): what they did, they did for the people to prepare for a New World Order. You start to question whether you're doing the right thing, and come to the conclusion that you are, because the NWO requires the sacrifice of free will, something heinous in the eyes of the Brotherhood. So you kill and kill some more, getting closer to that douchebag de Sable.
he discovers that de Sable has been planning something nefarious: to unite the Christian and Muslim armies together against the Assassins. Fuck. He chases down Robert and catches up to him in front of Richard the Lionheart (who is one fat motherfucker). Dick says a trial by combat should take place, with God deciding who is right. Altair wins, then tells Dick that God had nothing to do with it, he did it all himself (I wonder if he made a faces of atheism post about that. "I heard people praise God when I smote that bastard Robert de Sable. I DIDN'T SPEND MY WHOLE LIFE TRAINING FOR AN INVISIBLE SKY WIZARD TO GET THE CREDIT"). He then tries to convince King Dick that ending the war would be beneficial to both parties (Dick says no). Before he's proper dead, de Sable tells Altair that Al Mualim is actually a Templar too. What. A. Cunt.
you go back to Masyaf to do your own version of Star Wars (Dirt Wars?) and confront bossman. He tells you The Apple can conjure illusions, and that all religious miracles are products of the artifact. Altair's all "pfff, get da fuck outa here, you crazy fool", then Al Pooalim shows him some magic tricks, creating clones of himself. You then have a throwdown, and win. You recover the Apple from his body, and it turns on, showing a globe of the world with locations marked all over the place.
with the memory complete, Desmond is pulled out of the Animus to find Vidic chucking a conniption fit because a small band of Assasins are trying to mount a rescue effort to save you. Lucy, who you'd been talking with every time you came out of the Animus, tells you she's an assassin working undercover, and she's going to help you soon. When she leaves, you experience something weird: Eagle Vision. Walking around, you see symbols all over the floor and walls that you can't see normally. It turns out the previous person to do this shit, 'Subject 16', was in the Animus for so long he ended up crazier than a shithouse rat, and opened his veins up and painted the walls with his blood to leave a bunch of cryptic clues behind. After seeing enough symbols, credits roll