Why was Frodo so resistant to the ring?

Part of it was hobbit nature. The hobbits weren't included as a race that Sauron tried to convert to his power with rings. They were beneath his notice. There seemed to be something exceptional in the hobbits nature that really only Gandalf seemed to notice. They had a single minded simplistic view of life. This can be seen in their constant desire to not just eat but have a set schedule of eating times. So we have a race that is uncorrupted, basically good (there seems to be no crime in the Shire) and stubborn. Every hobbit seems content to live their whole life in the Shire and never venture outside except for Bilbo and his nephew Frodo. Even then Gandalf has to basically shanghai them to make them leave. The other hobbits that adventure with Frodo are brought along simply because they know enough that they would be tortured by Sauron and his minions to get the little bit of information that they do have. So their lives are already in danger.

Now Gollum was originally of a race that was very similar to the hobbits. Which helps to explain how he has lived so long and also becomes invisible when he wears the ring. The pull of the ring is not too great on the hobbits because it senses nothing remarkable or exceptional in them. The ring has a will of its own and will work itself towards evil anytime that it can. It takes a strong and stubborn mind to resist it. We see Boromir give in to the will of the ring and he's only near it. Gollum doesn't crave power. He only wants his precious. Bilbo does have cravings. He wants adventure and the ring latches onto that so that we see Bilbo has to give up his ring as well. The fact that he is able to is quite remarkable because he is the first possessor of the ring to willingly give it up. Frodo however has no real desire. He doesn't crave anything. He has what's described as a pure heart, but even he is affected by the ring. Sam is actually the only bearer of the ring that isn't affected by it. This is because he only desires to garden and help his friends. His lack of desire is displayed in his request for rope from the elves. He only debates giving Frodo back the ring because Frodo seems desperate for it as Gollum is.

Sam is actually considered the true hero of the story by Tolkien, and as Mistu4u pointed out, there is a lot of allegory and meaning he put into The Lord of the Rings. Peter Jackson did a lot to interpret his works as faithfully and truthfully as he could and also added elements that come from other books.

Gandalf and Aragorn were wise enough to know that the Ring's will would try to subvert them. Having a part of Sauron's soul inside it, it would convince people as it had Boromir and earlier Isildur that they could defeat Sauron with it -- that they could master it. The Ring was evil, and would always lead those that sought to master it towards evil. It was tricky, very tricky and would always seek its master or create a new Sauron (kind of like Voldemort in Harry Potter with his Horcruxes). Only one pure of heart and free of desire with a strong-willed mind could have carried the Ring without giving into its will. Frodo was the only choice.

Sam could have, but he didn't even have a desire to carry it to be destroyed. He would have tried to find a different hobbit as a way of dealing with it, probably similar to Bilbo's and Gollum's method -- keeping it close. Except Sam wouldn't have used it at all. He would have simply buried the Ring, which would then find a way to get to someone that it could work its will on. The time had come for it to be destroyed, and only Frodo was the right amount of brave, fearful and adventurous to get the job done.

I did not read the novels. I am answering from the movies (LOTR series) I saw.

When Elrond called the council of the races, they started fighting over what should be done to the ring. Boromir was allured by it almost instantly. It became clear to Gandalf and others within that sometimes one needs a person who is not greedy for power and who can not deterred from his aim easily. The ring had the power to corrupt a person. So taking the ring to the mountain was itself a very hard task to achieve. While fighting over the ring, Frodo was the one who stepped forward to volunteer in carrying the ring to the mountain where it was forged. Everybody then decided to let him carry the ring. There were several reasons behind giving him the responsibility.

1. Frodo had an innocent character. The innocence made him more immune to the ill-effects of the ring.

2. Frodo did not lust for power. Gandalf and others use power, so they can easily be deceived by it. But Frodo can not be, because he had no lust for power.

3. Besides, Frodo was successful in carrying the ring from the shire to the council. So he had also got some experience in carrying it. So others must have also thought that as until then Frodo was successful in carrying the ring to the council, maybe he had gained some immunity against the ring.

We saw that Sam could not withstand its power even for sometime, but Frodo could. This is because was Frodo's inherent nature to resist the power of ring whereas others could not. So we can say it was his nature which helped him to stand firm against the charms of the ring for longer time than others.

Maybe the context is not really that relevant, but I found a nice explanation in the yahoo answer by some author about why Frodo is not affected by the ring, from the writes point of view:

Tolkien, who wrote LotR, filled the book with a lot of Christian (Catholic) images. The threesome of Sam, Frodo and Gollum, who take the ring to Mordor, represent the three "parts" of man - spirit, soul and body. The Ring represents sin.

Frodo, who is the soul or mind, is the one who carries the ring. That is because that is where all moral decisions are made. So the soul bears the responsibility of dealing with sin. Gollum, as the body, is totally corrupted by the Ring. And in the end is destroyed by it. Sam, represents the spirit. That is why he is able to handle the Ring, and then returns it to Frodo without being corrupted. The reborn human spirit is not tempted by evil, nor will it sin.

Sam and Gollum are always in conflict with each other, just as the spirit and the body of most people are in conflict with each other. Poor Frodo, the mind or soul, is caught between them. Having to make compromises to bring the "body/Gollum" along, because he can not live with it. But wanted to be free of the influence of both Gollum and the Ring.

In the end, is the "Sword of the Spirit" (in Sam's hand) that cuts away the Ring, and the last of the flesh that still clung to it, and free Frodo from the influence of evil.

Excellent answers! One additional point is that Frodo, alone of all the ringbearers, was given the ring of free will. He did not acquire the ring by cheating or murder.

Frodo and Gandalf, in the chapter The Shadow of the Past, discuss how Gollum could not give up the ring, where Bilbo did:

‘What do you mean?’ said Frodo. ‘Surely the Ring was his Precious and the only thing he cared for? But if he hated it, why didn’t he get rid of it, or go away and leave it?’

‘You ought to begin to understand, Frodo, after all you have heard,’ said Gandalf. ‘He hated it and loved it, as he hated and loved himself. He could not get rid of it. He had no will left in the matter.

‘A Ring of Power looks after itself, Frodo. It may slip off treacherously, but its keeper never abandons it. At most he plays with the idea of handing it on to someone else’s care – and that only at an early stage, when it first begins to grip. But as far as I know Bilbo alone in history has ever gone beyond playing, and really done it. He needed all my help, too. And even so he would never have just forsaken it, or cast it aside. It was not Gollum, Frodo, but the Ring itself that decided things. The Ring left him.’

It might be helpful to start by recalling what was said about Tom Bombadil. When someone said that Bombadil had power over the ring, another person said that he would put it differently, that the ring had no power over Bombadil. Of all the races in middle earth, the hobbits are the most Bombadil-like. They don't crave power or beautiful things. No hobbit ever killed another hobbit in anger (or so it was said in the Scouring of the Shire). It is also worth recalling what Gandalf said about Bilbo, that he took so little hurt from the ring because the first thing he did when he possessed it was to spare Gollum's life. Perhaps Frodo possessed some of Bilbo's wisdom innately, and surely his own wisdom grew. He offers the ring to more than one person. Their reasons for refusing are helpful in understanding the ring's nature, but probably Frodo grew to understand the ring better himself. Also the fact that he was willing to give the ring up suggests that it had less control over him than it might have had.