One of my favorite games to play in a group is Settlers of Catan. For me, the fun part of the game is the balance between luck and skill. While the game seems to be primarily luck based, with players gathering resources based on arbitrary dice rolls, it really is deceptively skillful, with the most experienced player usually winning. In terms of mechanics, there are two dices used for the resource collection, which creates an interesting dynamic with probability; numbers closer to 7 are more probable, while 2s and 12s have only a ~3% chance each. Also, there are specific recipies to create points, so different resources have different priorities at different stages of the game. With the arrangement of the board, it can become easy to create a monopoly around certain resources, which allows players to interact more through trading arrangements, creating a sense of fellowship in some games.
There are also different goals in the game, such as making the longest road or collecting victory points, which will award bonus points. Thus, there is direct competition between the players, creating a more social dynamic. At the same time, because of the different ways to award games, players can express themselves in the way they play the game (for example, I usually pursue the longest road because I enjoy competing with other players and creating monopolies around resources).
In summary, the mechanics that make the game the most fun for me are the dice roll and number system for distributing the resources, and the necessity for adaptability with the different win conditions achievable with different resources.