Although I've played every edition of "Animal Crossing" that's been released so far, I'll specifically be sticking with New Life for the purposes of this list because it's the one with the most features and also my personal favorite game in the franchise!
1. Your Character's Birthday
Chances are that you gave your player the same birthday as you gave yourself, so that's a cool real-world parallel. All of the villagers in town, as well as your player's mom, will send you gifts and you get a birthday cake! Your favorite villager will even come and visit you personally to take you to their house for a surprise party!
2. Toy Day
Toy Day is the "Animal Crossing" version of Christmas, and it's pretty fun. You get to dress up as Santa Claus and give presents to the villagers, and then Jingle the Reindeer will appear and give you presents as well!
Pavé hosts the event in the Plaza. You get to play minigames and receive furniture! Also, the game is coded so that the weather will always be sunny on Festivale. It's a bit similar to real-world Mardi Gras, and it's great!
4. Bunny Day
Similar to Easter, Zipper T. Bunny will appear and give the character furniture in exchange for eggs that are hidden around your town. Personally, I like the egg-hunt game more than I like the appearance of the furniture itself, but it's the thought that counts. Also, Zipper T. Bunny, although kind, is a bit creepy. He's clearly another character in a fursuit, but we have no idea what character it is.
5. Weeding Day
Weeding Day is basically work disguised as a holiday, but it does have a decent number of redeeming factors. First of all, Leif will tell the player exactly how many weeds are left in town so that the player has an idea of how much work they have left to do. Second, you do get a prize at the end: flower furniture!
6. Fishing Tourney
The Fishing Tourney is hosted by Chip. The objective is to catch the biggest fish and/or a specific type of fish. It's a lot of fun the first few times, but sadly, it can get a bit repetitive after a while!
The Bug-Off is similar to the Fishing Tourney, except Nat hosts it and asks you to catch rare insects. The only reasons why I prefer the Fishing Tourney to the Bug-Off (slightly) is because 1. bugs creep me out and 2. the bug festival is a bit harder in my opinion.
8. April Fool's Day
On April 1st of each year, Blanca will appear and then disappear into one of the villager's houses and impersonate them. The player must find out who is real and who is fake.
Don't get the wrong idea from this one being the last on the list! I still really enjoy it, even though Blanca is creepy. However, my one major complaint lies in the fact that it's based almost completely on luck or prior knowledge, unless you cheat to look up the questions that the player/Blanca will ask you.
It doesn't feel as much like a "game" as the other festival minigames do. That being said, it is pretty cool that you can get the villagers' photographs if you guess correctly!
Although there are a few other in-game festivals, these are the ones that I think are most well known, as well as the ones that I participate in the most. There aren't any festivals I dislike, but there are some that I'd like more if certain things were added.