Beavers were a type of semi-aquatic rodent found in the world of Narnia. They were some of the most industrious, practical and social creatures in Narnia, known to be always busy with something, and were constantly found in charge of all manner of construction projects (like building dams for their homes). They were also known to have a network of useful holes and hideaways throughout the countryside.
Two Beavers helped the Pevensie children get to Aslan's Camp and help out in the First Battle of Beruna. In The Lion, the Witch and the Wardrobe, Mr Beaver is presumably the second Narnian that Lucy meets. He gives her the handkerchief that she originally gave to Mr. Tumnus before he was arrested by the chief of police, Maugrim, on behalf of the White Witch.
Mr. and Mrs. Beaver then continued to help the Pevensie children to fulfill the Golden Age Prophecy of when two sons of Adam and two daughters of Eve would bring about the downfall of the White Witch. The children set out on a perilous journey to find Aslan with the help of these two loyal Beavers.
The first claim is untrue, since the Beavers were among those given "rewards and honours" after the Pevensies' coronation, however Beavers do not appear in the books again until the He-Beaver and She-Beaver are seen in Aslan's Country after Narnia's destruction. This must mean that the second claim is true, and the Beavers must have died at some point during the Telmarine occupation.