This writeup concerns only the book by J.R.R. Tolkien and not the film (which I have not seen). Reading this node may ruin the book for you, as I make no attempt to keep any secrets.
The Fellowship of the Ring continues the story that Tolkien began in The Hobbit. It seems that the ring that Bilbo Baggins took from Gollum was far more important than he had ever imagined. The story begins in the shire many years after the events of The Hobbit.
The first part of the book concerns itself with the events surrounding a birthday party that Bilbo was throwing for both himself and his nephew Frodo. Bilbo has this grand plan of vanishing at the high point of the festivities, and then going off and leaving the shire for good. Bilbo plans his party (and vanishing act) for a long time in advance, and his party proves to be one of the grandest the shire has ever seen. Bilbo goes through with his plans to vanish at the end of his birthday speech, but Gandalf throws in a little bit of magic to make the disappearance seem a bit more like trickery than magic.
Bilbo leaves his ring behind with his nephew Frodo, along with many of his other belongings. Many years pass by and Bilbo still has not returned from his journey (don't worry hobbit friends, he is safe). But Gandalf makes an appearance to let Frodo know that the ring is far more dangerous than they had thought, and that the evil Sauron is searching the whole of Middle Earth looking for it. So Frodo sells Bag End (his hobbit hole, that once belonged to Bilbo) and sets out with Sam Gamgee, Merry Brandybuck, and Pippin Took in hopes of reaching Rivendell. There they hope to find out what they should do from there.
Hobbits being hobbits they quickly get themselves into all kinds of trouble. While traveling through the woods they encounter a dangerous willow tree that attempts to trap them. They are nearly defeated by the tree before Tom Bombadil comes along and frees them. The group stays at Tom's house in the forest and then sets off again.
Our intrepid hobbits soon make their way to the city of Bree. While in Bree they stay at an Inn where they meet Strider who is both a friend of Gandalf and a ranger. They also receive a letter from Gandalf from the innkeeper. The letter warns them that they should leave on their journey immediately, and was dated several months prior. Before leaving the inn Frodo manages to use the ring's power in full view of everyone, and the next morning they find that all their ponies have been stolen (revealed themselves to the enemy it appears).
They soon set out again, this time with Strider leading the way. Black riders who are looking for the ring pursue them constantly. Eventually one of these riders manages to wound Frodo with his sword, injuring him gravely with a wound that isn't just physical. With the help of some elves they manage to temporarily defeat the black riders while crossing a river. The riders are all swept away by a mighty current, and Frodo passes out from his wounds.
Frodo awakens in Rivendell where he learns that Bilbo is indeed still alive and has been living there for quite some time. It is here that Gandalf finally shows himself again. A great council is convened, and they decide that the ring must be destroyed. But doing that will require carrying the ring into Mordor, where the dreaded enemy Sauron has power over the land. They assemble a group of nine (The Fellowship of the Ring) to carry the ring to Mordor. In addition to the hobbits already mentioned the group includes Gandalf, Aragorn (who was known earlier as Strider), Gimli, Legolas, and Boromir. Before they leave Bilbo gives Frodo some of his old adventuring gear, including his mithril coat and his magical sword "Sting" (which glows in the presence of humanoids).
It is winter before the group finally leaves. They make an attempt to pass through the mountains, but it quickly becomes apparent that they will be unable to take that path. So instead they decide (after much debate) to go underneath the mountains, and through the mines of Moria. They travel the fourty mile distance through the mines mostly in the dark. Eventually they end up being pursued by minions of Sauron. They manage to escape, but only because Gandalf sacrifices himself for their safety. Gandalf falls into a pit, so it is still possible that he may be alive, but this book does not let the reader know one way or the other. But all the characters that where with him assume that he perished.
From Moria they go onto Lothlorien, which is an ancient elven forest. They meet up with Galadriel, who is a great elvish leader. She provides them with much information, many magical gifts, and supplies of a more mundane nature as well. She even gives everyone elven cloaks to help them blend into the forest more easily. It is here that they great rift between elves and dwarves begins to mend, as the elves accept Gimli, despite the long standing rules against dwarves in Lothlorien.
The fellowship leaves Lothlorien in several small boats (which I believe were canoes, but the book only seems to describe the wood they were made out of, and not the boats themselves). After a while they learn that they are being pursued by Gollum and that the armies of Sauron are not far behind. After a narrow escape from the enemy, Frodo goes off alone to decide what to do from that point on. Boromir appears and betrays him, attempting to take the ring for himself. Frodo decides that it would be best to finish the journey alone, and he attempts to do just that. But Sam catches him at the last second and convinces Frodo to let him come along.
This is where the book ends. This tale is continued in The Two Towers.
I really enjoyed this novel. I have but one bit of advice for prospective readers. Don't try to read this book all in one sitting like I did. Tolkien loves describing landscapes so much that you will end up not seeing the hobbits for the trees (so to speak). I suggest reading only a single chapter a day, it will be much more enjoyable that way.