I've heard a lot about the Undying Lands but I don't fully understand how they're different from Middle-earth and why only certain people/races are allowed to go there.
Can someone explain?
This is a pretty broad question (in the realms of Tolkien it's roughly equivalent of asking "What's Africa?") but here goes...
The Undying Lands (or Aman, or the Blessed Lands, or Valinor) is a continent that was to the west of the Middle Earth we see in the Lord of the Rings. It was home to the Valar, Maiar and most of the Elves throughout the First and Second Ages.
As part of their reward for aiding the Elves in the War of Wrath against Morgoth, the Numenorians were gifted an island from which Aman could be observed from its highest peak - Numenor. This set-up worked well until the Numenorians became prideful and firstly conquered and captured Sauron, bringing him back to Numenor, and then, under Sauron's urging, attempted to land on Aman with a fleet.
This prompted the Valar to appeal to Eru to act to preserve the sanctity of the Undying Lands. Eru removed the Undying Lands from the earth - converting the earth to a sphere in the process (it was previously flat in Tolkien's mythology). Eru also sank Numenor, killing the majority of them save for the Faithful - Elendil and his sons who then founded Gondor.
From that point through the Third and Fourth Ages Elves were allowed to continue travelling "home" to Aman, or "heading west" as it is referred to in the Lord of the Rings a few times. The Straight Road was left open for them, and only them, to travel (see my answer here for more details, as well as on the exceptions).
If you wish to know more, I would definitely recommend the Silmarillion for the history of Middle Earth prior to the end of the Third Age, which was when the Lord of the Rings was set.