For me as a 13 year old playing the original Animal Crossing on a gamecube, it was a harmless escape from life. At the time I was dealing with the loss of my older brother and was very depressed.
Animal Crossing is a simulation game sort of like the Sims (not the city building sims) or Harvest Moon. The difference is the time in Animal Crossing is the same as in real life. On Christmas if you turn on Animal Crossing, the town in the game is celebrating Christmas (well actually "Toy Day" to be politically correct). The game followed real life seasons as well. This makes playing the game for a little each day rewarding as there is something new every day for at least a year. My favorite aspect of Animal Crossing is how there is no absolute goal or achievement you strive for. The game is about doing what you feel like at your own pace. The soundtrack is one of the most relaxing soundtracks of any game I have ever played and the tunes change by the hour. For example, you can get lost in the game for hours fishing trying to catch the rarest fish in the game. This game was therapeutic for me as I was young and dealing with some rough stuff in real life.
If you're looking for a more fully-featured version of Animal Crossing, check out Animal Crossing: New Leaf for the 3DS. It's definitely the best entry in the series. It even has online multiplayer with random matching and text chat! (A rarity for Nintendo.) If you want to pirate it on 3DS, there are ways, or if you want to play it on PC, it apparently works alright in Citra, the 3DS emulator.
If you don't have a 3DS, Animal Crossing: Wild World is available for the DS and consequently its emulators. (The best DS emulator is probably DraStic for Android.) If you have a Wii, you can try Animal Crossing: City Folk or the original, either legitimately or via modding.
Do note that while it is a "town simulation" you are playing as an inhabitant of the town (and in New Leaf, its mayor). If it sounds interesting to you, watch a YouTube video or two. It's incredibly relaxing.
It's a town simulation game that makes use of the real clock. So different things happen depending on the time of day you are playing or the day of the year for special holidays and events. There is only so much you can do per day, so it encourages playing every day to collect new items, maintain the town, and make money to pay off your house.
Yes! Fun fact: the GameCube version is almost a straight port - the whole game fits in memory, so you can take the disc out of the console after you start and it'll work fine.