Undying Lands was a name for Aman, or at least that part of it inhabited by the Valar, Maiar and Elves. The island of Tol Eressëa is several times identified as the easternmost of the Undying Lands, and, at the least, Valinor must also be included.
"Undying Lands" seems to be a name that originated among Men. The Númenóreans, especially, envied the seemingly endless life of those who lived in these regions. From the first, the Valar placed a ban on the Men of Númenor, that they should not sail into the West from their island, or set foot on the shores of Aman.
Wise as the Valar were, though, they did not foresee the wiles of Sauron. The fallen Maia falsely persuaded the last King of Númenor, Ar-Pharazôn, that the ruler of the Undying Lands would be undying himself. Believing Sauron, Ar-Pharazôn assembled a great navy and sailed westward to make hopeless war on the Valar for the imagined prize of endless life.
The Valar could not permit this: Manwë called upon Ilúvatar, and the land of Númenor was destroyed and lost forever. The Undying Lands, which until that time had been part of the World, were removed forever from the reach of mortal Men, though the Elves could still sail West and come there, if they would.
It is to the Undying Lands that the White Ship sails at the end of The Lord of the Rings. The Ring-bearers, Bilbo and Frodo, were among the very few mortal beings to set foot on the shores of the Undying Lands. Tolkien is careful to point out, though, that even in Aman mortals remain mortal.