It's a life simulator set in real time.
You get loaned a house, or in this case an island, and you get to customise it as you see fit. In the latest game, you get to plan the entire neighbourhood including the location of shops, villagers, rivers and walk-ways.
The reason why everyone loves it because there's multiplayer aspects. Like Pokemon, each town has their own exclusive items, so you're encouraged to visit different islands to collect these items or discover villagers so you can invite them to your town.
It's not a game for everybody. The game doesn't really have an objective, it's entirely up to you and your friends to play around with. However, it's cozy and it's the type of game that you can just pop in and you'll find there's been all sorts of developments since your last playthrough.