The difficulties with trilogies is that while they are three books the same plot and theme tends to run through all three of them. It really does depend on the story though as there are a number which are self contained, but due to the popularity, the author writes a second, third (and sometimes more). However, this is not the case with Lord of the Rings. Tolkein originally intended that the story be in one book, however for economic reasons it was decided that it would be released as a trilogy.
The plot (which you probably already know anyway) pretty much runs through all three books. The hero, a Hobbit named Frodo Baggins, inherits a ring which turns out to be a powerful artifact that the dark lord Sauron wants back. However, very bad things will happen if it falls into the hands of Sauron and it is decided that the ring must be destroyed, but the only way to do it is to throw it into the 'Cracks of Doom', a volcano where the ring was originally forged (that also happens to lie in the middle of Sauron's domain).
Now, I will spend some time talking about the nature of the ring, and also look at the history of Middle Earth as it comes up to the Lord of the Rings. Lord of the Rings is unique in that the main antagonist is not so much Sauron, but the ring. The ring represents power, and whoever possesses the ring wields great power. However the ring has a very corrupting influence over all of its owners, and one cannot possess the ring without becoming mad, or even coming under the influence of Sauron. This is what makes the ring such an awesome antagonist in that it cannot be used for good, no matter how noble the intentions of the owner are. It will always corrupt you.
Frodo is not immune to this, and we can see this at the end of the story when, despite all of the hardships that he endures to reach his goal, he simply cannot let go of the ring. It is only by accident that the ring falls into the Cracks of Doom and is destroyed. Secondly the ring is a burden, and we are constantly reminded on how it weighs down Frodo's soul. He does not want the burden, but he knows that it has fallen to him to carry it, but even then, despite him being pure of heart, the ring will and does corrupt him. This is such that despite completing his quest, he simply cannot return to his normal life in the Shire and must leave with his uncle Bilbo Baggins to travel over the seas with the elves.
When Tolkien originally drafted the books he did not do so in isolation. Many fantasy novels (and films) are simply created out of a vacuum. Nothing really existed before and nothing exists after. All we see is what occurs in the action of the story. Granted there might be a rough history that leads up to the events of the story, but nothing all that deep. This is not the case with the Lord of the Rings. When Tolkein drafted his story he did it against a very deep and rich background. His Middle Earth is not a trilogy about a hobbit and a ring, but a living and vibrant world with a deep and colourful history.
Granted, not everything in the history of Middle Earth is important to the plot, but it adds character and colour to the story. The actual story begins at the end of the second age when the king of men, Isildur, goes to war against the forces of Sauron, and while Sauron is defeated, Isildur does not have the last laugh. He removes the ring from Sauron's hand and is immediately corrupted by it, and then finds himself killed in an ambush at Gladden Feilds. That was 3000 years before the current events, and there is a lot that happens before and between them as well. This history is outlined in the appendix to Return of the King for those who are interested, and there are numerous other books containing tales from the history of Middle Earth (the best being the Silmarillion).
However, I will finish off with Sauron. Sauron is the bad guy, but in the grand scheme of Middle Earth he is not the bad guy. That title falls to his boss, Morgoth. Now, Morgoth vanishes at the end of the First Age when Belariand is destroyed and the world remade. However, Sauron is his leiutenant, and when Morgoth steps out of the picture, Sauron enters. He is a subtle and manipulative individual, and while we do not meet him in The Lord of the Rings (he always floats in the background, and we know that he is present and that he is powerful, but he operates through his minions such as Sauruman and the Nazgul) he does make appearances in the earlier story. What comes to mind is the tale of Numenor, which is Tolkien's version of Atlantis (more on that later). Numenor was a mighty and powerful human empire, but they were corrupted, and this came about because they allowed Sauron to become their king. It is not that Sauron is openly explicity but that he goads others to commit evil acts, and he does this with the Numenorians when he goads them into sailing west to take on the Valar (sort of very powerful angelic beings). It is not Sauron that is punished though, it is the Numenorians as their empire is destroyed and they are scattered to the four winds.