A Link Between Worlds is about as close to a sequel as The Legend of Zelda has ever received. Often we'll see allusions to the previous Hero of Hyrule, a blonde boy in green, saving a previous Princess Zelda (apparently the royal name carries a bit of a curse) — but A Link Between Worlds' Hyrule is the same Hyrule as A Link to the Past. The same map, the same races, the same landmarks. It's actually set several hundred years after A Link to the Past (we'll abbreviate as ALBW and ALttP hereon). Ganon is long gone — but the game begins with our hero stumbling upon a new menace: a gender-ambiguous villain called Yuga who is turning people into paintings and nicking off with them.
Of course, fans know what happens next: you have to complete dungeons to collect the three pendants of courage, power and wisdom that will allow you to claim the master sword, followed by an in-depth adventure between two worlds to rescue the seven sages and defeat the game's Big Bad. In ALttP, the overworld was shadowed by the Dark World; in ALBW, Yuga's shenanigans have opened rifts in Hyrule to a mirror version of Hyrule called Lorule, presided over by the dark-haired Princess Hilda.
This is part of what gives the game its name. The other is Link's means of getting to Lorule: his new power. Using a mysterious bracelet, Link can slide into walls and sneak along in two dimensions; so not only is he between Hyrule and Lorule, he is between the 3D world and the 2D.
As an aside, this is tremendously effective when you have the 3DS's 3D capability switched on, and trust me — you'll want to at least try this. This is the first game I've seen on the 3DS that uses the device's stereoscopic 3D to good effect. It has a genuine sense of depth without being dizzying or headache-inducing, like so many games. It's actually the first game I've played on the console where I've kept the 3D turned on the entire time, just because it looks and feels amazing.