I want that special edition, but alas, it is not meant to be (yet!).
A basic gestalt on the game ? I have no Fking idea what is supposed to be.
I guess at its most basic, Animal Crossing is a life sim.
You make a (human) character and then live in a real-time virtual town where you meet neighbors (anthropomorphic animals), design your house, interact with other players, go fishing, catch bugs, and collect. There's no real goal or endgame. In the short term, the closest to a official goal would be paying off loans to upgrade your house. In the long term, there's really nothing beyond what you decide: collecting all the furniture (there is a ton), filling out your bug or fish catalog, perfecting the design of your town, and so forth.
Although time traveling is common (by setting the game's clock ahead), the game is meant to be played in real time. If you pay or reach the requirements for your house, museum, or store, et cetera, to be upgraded, you won't see it until tomorrow. Holidays and festivals roughly analogous to real world holidays will occur on or around the same days in-game as they do in real life. Animals will move in and out, and sometimes special animals will show up with rare goods, little events to participate in, and so forth. Certain bugs and fish are only available during certain hours of the day and certain months/seasons of the year (again, realtime, and based on your real-life hemisphere).
In older games, you could have up to four players in one town, but you had to take turns. More recent games feature concurrent multiplayer and online play, with New Horizons allowing four players at once locally, and up to eight online.
Also new to New Horizons, there's a very light bit of survival added to it, with tools that constantly break and need repairs, the ability to craft furniture, and some basic terraforming aspects. You can basically set your island up exactly as you want it, including the placement of the houses and buildings.
TL;DR: you live peacefully with animals, furnish a house, build up a town with amenities, and collect stuff, all in real time.