Fate Stay Night is a visual novel with three very different stories that start the same, but diverge early on. Some of the characters and story points aren't developed until the second or third route. Thus far there have been two anime series made. The TV series mostly follows the Fate story while cherry picking some parts of the other two. The second story, Unlimited Blade Works; is covered in the film of the same name. Heaven's Feel hasn't been done as an anime, but it's unlikely it ever will - it's honestly too dark to do properly.
There is one ending for Fate, and two endings for the other two. I split the end into their own respective pages to avoid spoilers.
Fate Stay Night
It's been 11 years since a large fire consumed a portion of Fuyuki city. Shiro was just a child when his parents were killed. He was taken in by a man named Kiritsugu Emiya. Kiritsugu could wield magic, and passed on a few of the basics to Shiro. The night before Kiritsugu passed away, he confessed that he never lived up to his ideal of a hero. Shiro swore that he would become one and fullfill Kiritsugu's dream. In the present Shiro is much like any other guy in high school, aside from practicing magic in secret each night.
A girl named Sakura, one year his junior; often comes to his house to prepare breakfast and dinner. It's been a routine for about a year, but as Sakura has matured, it's starting to trouble him. The days pass with little changing until one night Shiro witnesses a battle between two god like beings. Having seen something he shouldn't have, one of them seeks him out and kills him. Or nearly kills him. Shiro was indeed mortally wounded and lost consciousness, but somehow survives with his wounds healed when he wakes up. He heads home only to again encounter his would be assassin. Things look hopeless, but in a flash of light a girl dressed as a white knight appears and saves him. She drives off the spirit before informing Shiro he is now her master. Her name is Saber.
Saber pursues the spirit outside Shiro's house. Shiro follows her. The spirit escaped, but there is another right outside. Saber quickly dispatches the servant and is about to kill a person nearby when Shiro stops her. As the person steps into the light, it turns out he saved the beutiful school idol Rin Tohsaka. She's normally distant and rarely speaks much, yet she greets Shiro casually and lets herself into his house. Is this the same Rin Tohsaka?
Rin explains she, and now Shiro; are masters. There are seven such people (normally magic users) who participate in a contest to obtain the holy grail: a vessel which can grant a wish. Each master has a heroic spirit called a Servant to fight for them. There are seven servants, and each has a different class. A master has three command spells to control them. Fights in front of normal people are prohibited. Anyone who doesn't want to fight anymore, can take refuge at the church. Aside from that, there are no rules.
The situation is troubling for everyone. Rin was interested that Shiro has become a master, even summoning the servant she had intended; only to learn that he's an armature that lacks even the basic skills. Saber is also concerned that the master who summoned her is hardly even a magic user. Shiro is confused as a girl in armor claims to be his servant, and the school idol is in his house babbling about a contest to the death. Life is sure going to get interesting.
Shiro has the choice to back out of the war if he wants. He doesn't want to harm anyone, and certainly not for a grail he doesn't care about. However he's already been a victim and seen the potential for casualties in this war. If he's been trying to become a hero who can save and protect people, then this is precisely the opportunity he's wanted. He also owes Saber for saving his life.
Shiro confirms he will be a master at the church. Even if he doesn't wish to possess the grail, he has other reasons to fight. His goal isn't an object, it's an ideal. He'll fulfill his promise to Kiritsugu.
Shiro starts walking home when he encounters Illyasviel von Einzbern: a master of the grail war. Her servant is Berzerker. Her intent is to kill him. Shiro's choice to take part in the grail war is only the beginning. He'll have to make more tough choices, many of which will have major implications on the course of the grail war, and what will be required of him to become a hero.