UPDATE: See below for a discussion of the Eagles’ role in The Hobbit films.
One of the most commonly asked questions about The Lord of the Rings is why the Fellowship didn’t just fly into Mordor on the Eagles. It is frequently illustrated by the (in)famous How It Should Have Ended video:
The video is, of course, comedy, and not all Eagle scenarios are the same, yet it raises an interesting question. However, the reason the Ringbearer did not fly to Mordor by Eagle is fairly simple: the purpose of the Fellowship of the Ring and the linchpin of the entire strategy decided on in Rivendell was to destroy the Ring in a mission of secrecy.
Taking the Eagles might have worked, I will grant that. It may have been a successful mission and allowed the Ring to be destroyed earlier than it “actually” was. However, it would have sacrificed secrecy and drastically increased the changes of the Ring being captured. When you have the fate of the world hanging in the balance, you don’t want to take any unnecessary chances.
Eagles are, clearly, far more noticeable than Hobbits or other travelers on foot. We don’t know how exactly Gandalf planned to get into Mordor, but we can surmise that they would have gone through a mountain pass or valley some where. We know of only three (the Morannon, Cirith Ungol, and the Nameless Pass), but it stands to reason that there were more. Not ideal ones, perhaps (though Cirith Ungol itself was not ideal), but mountains are not impenetrable and continuous walls of rock.
An Eagle flying through the air would be easily noticed by Orcs or other watchers (remember the sinister and sorcerous ones at the Tower of Cirith Ungol – there might have been more). Travelers on foot could sneak around much more easily, scout ahead (especially with a ranger), and slip by unnoticed (remember how quiet hobbits are?). The Eagles might have been able to slip by unnoticed, but it would have become far more likely that they would have been caught. Once inside Mordor (if they even make it), there is still the chance that the Eagles could be caught. There are the threats of the Nazgul’s fell beasts, and archers (the Eagles in The Hobbit were afraid of shepherds with bows, so one can imagine how they might react to trained soldiers).
The “classic” Eagle plan, as outlined in the YouTube video, would not work for a couple of reasons. First, the Ring could not just be dropped into the caldera; it had to be taken into the Crack of Doom itself: the center of Sauron’s sorcererous powers. The Crack of Doom was at the end of a tunnel that bored into the mountainside, and an Eagle would likely not fit inside, so it would have to bring have a rider. This would limit the height to which it could fly (the rider would need to breathe) and its agility during a fight. Yet more possibilities for failure. Second, a giant Eagle landing on the slope of Mount Doom would be quickly evident to any troops stationed there. A small group of people on foot might be able to sneak up unnoticed. Again, the Eagle plan might work, but it increases the chances of being caught.
In conclusion, the Council of Elrond did not know exactly what to expect in Mordor, so they had to plan for the worst (i.e., assume the worst case scenario for each possible solution). The Fellowship plan was itself a very long shot and indeed, it failed in its original conception, though obviously a fragment of the Fellowship persisted. The Eagle plan raises such a host of potential issues and problems that I think it is quite understandable why the Council opted to send people on foot. As I mentioned at the beginning, their concern was stealth, not speed.
Update: A commenter has asked why the Eagles in The Hobbit: An Unexpected Journey did not simply fly the Company of Dwarves all the way to the Lonely Mountain, particularly since the mountain was visible at the end of the film. The situation in The Hobbit is complicated by certain fairly minor changes made by Peter Jackson to the story. In the original novel, the Eagles agree to help Gandalf as a favor, in return for him having saved the life of their chief before the events of the story. However, they were afraid of the men of the Vales of Anduin, who shot arrows at the Eagles (including the chief whom Gandalf saved) to keep them from stealing livestock. Therefore, the Eagles took the Company only a short distance.
The situation is muddled in the film because the Eagles do not speak at all, which removes the explanation of their motives. (It is possible this will change in the Extended Edition.) The final shot of the Company looking towards the Lonely Mountain is also misleading in its depiction of the distance left before them. The Eagles drop the Dwarves off on top of the Carrock, a large rock in the middle of the River Anduin. The Anduin flowed through a broad valley to the east of the Misty Mountains, placing the Carrock nearly 200 miles away from the Lonely Mountain: a far longer distance than the Eagles were willing to fly simply to repay a favor. The Lonely Mountain would not realistically have been visible from the Carrock, but Peter Jackson chose to show it for dramatic reasons.