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