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.
The new power, like any good item or power in a Legend of Zelda game, will require you to think around a problem from every possible angle. If there's a place you can't get to or a task you can't complete, there's usually a combination of items that are required. For example, in the Desert Palace, you need the Sand Rod, a magic wand that creates walls/walkways out of sand. Sometimes, the walls are too tall to walk over — in which case you need to merge with the sand wall and find your way around it before it crumbles.
And, of course, each time you manage to figure a puzzle out, you get that delicious "Ohhhhh!" moment of cleverness.
The merging power is not the only thing that's new. The character who gives you the wall-merging bracelet is a mysterious, garrulous, bunny-masked chap by the name of Ravio, who makes it a gift to you when you let him stay in your house. While you're away adventuring, Ravio decides to rearrange things — turning your humble hut into his new place of business, an item-renting store. The items he has to rent are the tried-and-true Zelda mainstays: bow, bombs, hookshot, boomerang, ice rod, fire rod, wind rod and sand rod.
This means that, as soon as you have the rupees, you can rent out every single item from the beginning of the game, carrying around a decent arsenal. With one caveat: if you die, the items will be returned to Ravio and you'll have to rent them all over again. Since this is around 50 rupees each, it doesn't feel like a big deal — but eventually Ravio will allow you to buy each item, for a minimum of 800 rupees each.
You'll want to do this, and the reason why ties into a collection quest. A peculiar giant octopus called Mother Maimai has lost her 100 babies. For each 10 you find and return to her (hiding on walls, in trees, under tiles and rocks, in sand), she will upgrade one of your items — but only the items you own. It's such a classically convoluted Zelda mechanic: you're offered an incentive not to die, because you need to save your rupees to upgrade your gear, which you can only do by embarking on a long collection side-quest — which in turn encourages you to explore, sandbox-style, leaving no stone on the game's map unturned.
It's hard for me to predict how someone who has never played a Legend of Zelda title will react to a Link Between Worlds. It's a different time now. I would like to think that even in the current climate of GTA and Mass Effect and Skyrim, the same sense of vastness, discovery and wonder are still present in A Link Between Worlds in the same way that they were in A Link to the Past — that a top-down game can still feel like an incredibly rich adventure.
As a fan of the series for nearly 30 years, though, I can tell you exactly how it feels: it feels like coming home.