I have loved Animal Crossing since the GameCube version first appeared in 2004, and it's easily one of my favorite game series of all time. I've played all of the versions, for months and months on end, and I think there are a number of reasons why Animal Crossing ends up being so addictive.
Open-Endedness - First of all, I think Animal Crossing (and especially New Leaf) draws people in because of its open world style. You control how your town turns out, how your character dresses, the house you build, and what you do in the game. Feel like deep sea diving? You can do that. Want to capture all of the bugs in the game? That's an option too. Into landscaping? That exists too. You can even fish in a ballerina outfit
World Persistence - One of the major appealing points of the game is the persistent world, which exists whether you are in it or not. Time passes on par with the real world, with new content introduced each day in the form of fossils to dig up, items to purchase, and more. If you neglect the game for an extended period of time, your animal villagers will notice and become concerned.
Variety - There's always something to do in your village, which can keep you occupied for fifteen minutes or five hours. There are bugs, fish, and sea creatures to catalog, items to collect, badges to earn, and island mini games to complete. There are also special visitors, like Sahara the wallpaper saleswoman and Katrina the fortuneteller who come on random days. Items in stores are refreshed on a daily basis, trees regrow, and flowers spawn, which means players are always encouraged to log in once per day. You can even get take out coffee to drink on the go!
AI - The artificial intelligence in the game is and always has been fantastically entertaining. Villagers have unique personalities, phrases, and interact with the player in constantly changing ways. As you develop friendships with your animals, you will unlock new interactions with them. The changes that you make in your town can also influence how your villagers react to you, as can items that you're carrying and tools you have equipped. Earning the friendship of my villagers and speaking with them to hear their cute and quirky sayings has always been a favorite part of the game for me. Villagers will also make requests of the player, asking for items, to visit the player's house, or to play games like hide and seek, which increases the number of things that can be done in the game.
Content Rotation - Though Animal Crossing utilizes an open world concept, there are continual events that only happen during certain times of the year. There are fish and bugs that can only be captured in specific months and events that only happen when they happen in the real world. For example, during the summer, there are bug hunting competitions each weekend, while that might shift to fishing in the spring. There are also specific holidays, such as Bunny Day or Valentine's Day, where the player can get special items that can't be obtained any other time. This leads to not only logging in at least once per day, but logging in once per day all year long.
Gender Neutrality - Animal Crossing New Leaf is highly gender neutral. While it might skew towards a female audience given the laid back gameplay and the overall cute factor, it contains just as many gameplay elements geared towards a male audience as well. It's also fairly age neutral, being suitable for children but entertaining enough for adults to play. In fact, I only know adults that play.
Multiplayer Gameplay - Animal Crossing has always been fun with more than one person, and New Leaf adds to that by allowing players to play mini games together on the game's island, which can be a lot of fun. It's also fun to meet up with people, explore their towns, and speak with their villagers.
Engagement - Because there are so many things to do and so many different elements to the game, it all manages to feel completely engrossing. Animal Crossing doesn't feel like a video game, it feels like a tiny world that lives inside the Nintendo 3DS. I (and I'm sure many others) develop an emotional connection with the game, the tiny town that I run, and my villagers, which encourages me to keep checking up on the game day after day.
Of course, all of these gameplay elements assume that you don't use time traveling to earn in game achievements earlier than expected. The whole point of the game is to play a bit each day, watching your village slowly grow and progress over the course of a year or more.