A lot depends on what "style" of game you want to play....
Monopoly is an "elimination" style game. You keep playing until only 1 player is left. This is often classed as a "US" style game. While they can be good fun, it can leave people kicked out of the game earlier, and not having much fun as they watch the remaining players carry on playing. Monopoly, Twilight Imperium are these types of games.
Another classic style of game is the "Euro" game (been around for a while, but seen as a style originating in Germany). These games do not eliminate players, and allow all players to play through to the end. Normally, there is still a chance (though it may be pretty faint!!) for all players to have a chance of winning, or at the least, influencing who DOES win, keeping interest alive in the game to the end (Settlers of Catan, Carcassone, Alien Frontiers)
The other major type of game is a "co-op", where the players play together against the game (Pandemic, Pandemic Legacy, Gloomhaven (best. game. ever.), Battlestar Galactica etc. Some of these can be a bit more complex by the nature of the game - to make it a tricky opponent to beat, the rules can sometimes be a bit complex
Overall, I do like the US style games for 2-player fun - my partner and I do well playing against each other.... but no so well playing co-op (we consider the fact that we played the computer game "Portal 2" in co-op mode to the end, and our marriage survived, to be a major success!!). For more than 2 players though, especially for family games, we prefer the Euro style games.
... and yes, I am a board game geek.
Speaking of which:-
most popular board games
Have a look at the above :-)