"Modern” board games, aka “Eurogames”, generally have easy to understand rules, game mechanics which don’t rely much on luck, have quite a bit of interaction between the players and most players have a chance of winning for most of the game. In contrast the better known games such as Monopoly and Risk rely heavily on luck, have poor game mechanics and grind players down to the point where it is clear that they have no chance of winning but are still forced to play on, seemingly for hours, until they are eliminated or they quit. Monopoly may be the most popular boardgame on the planet but it's a poor game in comparison with the vast majority of EuroGames.
The bit about the rules is important; many boardgames designed in the 1970s and 1980s have lengthy and often hard-to-follow rulebooks (e.g. most wargames from this era). Modern games have short rulebooks which don’t read like a legal textbook.
https://en.wikipedia.org/wiki/EurogameBest game companies: If a game is made by Days of Wonder, Gale Force Nine, Ravensburger or Rio Grande Games, the chances are that it’s a pretty good game (most of our group’s games come from these four companies).
We have played Carcassonne. From what I remember it was pretty good, but Settlers of Catan appeared soon afterwards and took over the group. Here’s what our group of four players prefers in rough order of popularity / frequency of play (most of these games have their own Wikipedia page):
Games which play well with two players, but also play well with more playersTerraforming Mars – build colonies on Mars and make the planet habitable over many generations. Can be played as a solo game. In the year before lockdown, we played this more than anything else. There isn’t too much player interaction, but we found that given all the possible combinations of project and event cards in the game players were too busy thinking about their own position to worry much about the other players (especially when you add the expansion sets which introduce colonies on many moons within the Solar System). Like many EuroGames, it is a race to collect the most victory points by game end. We have experimented with allowing players to trade and sell project and event cards with each other, but this greatly slowed down the game.
https://en.wikipedia.org/wiki/Terraforming_Mars_(board_game)Small World – players run tribes in a fantasy “Small World”, exploring and fighting with their neighbours. Lots of expansion sets.
https://en.wikipedia.org/wiki/Small_World_(board_game)Ticket to Ride – travelling on the railways, score victory points for completing routes. Arguably the easiest game to learn on this list. There are many versions, including super quick games that usually take no more than 15 minutes. Start with the original Ticket to Ride, which covers North America.
https://en.wikipedia.org/wiki/Ticket_to_Ride_(board_game)Dominion – deck building card game, set in a medieval world. It’s really a race to build your deck and acquire victory cards, but there isn't much interaction with the other players. There are a lot of expansion sets for Dominion.
https://en.wikipedia.org/wiki/Dominion_(card_game)Caylus – build a medieval town and castle. This is a worker placement game; put your “men” on spaces on the board, such as the builder, and then use that space to do something (doing so excludes others from using that space this turn – the race to use certain spaces during the turn is a key mechanic in games like this)
https://en.wikipedia.org/wiki/Caylus_(game)Ys – Another worker placement game, this time it concerns controlling parts of a port-city.
Other games. These can be played by two, but they are much better games with more players:Settlers of Catan – establishing settlements on the newly discovered island of Catan. Produce resource, trade these with the other players, use them to build settlements, cities and roads. Settlers has lots of expansions and is extremely popular amongst Eurogamers.
https://en.wikipedia.org/wiki/CatanFive Tribes – Five Middle Eastern tribes trade and explore, whilst in some cases summoning Djinn / Genies to help them. Another worker placement game, this one has the theme of 1,001 Arabian nights. There are plenty of ways to stitch up, I mean "interact with" your opponents in this game. Of all the games I've played where players collect victory points to determine the winner, this is the one with the most variety of ways in which to collect points.
https://en.wikipedia.org/wiki/Five_Tribes_(board_game)Puerto Rico – Colonial Governors importing workers to grow crops, build buildings and export goods from the island of Puerto Rico. For many years Puerto Rico was the top rated game amongst Eurogamers.
https://en.wikipedia.org/wiki/Puerto_Rico_(board_game)Thurn und Taxis – building postal networks in 16th century Bavaria and its surroundings. Very easy to learn.
https://en.wikipedia.org/wiki/Thurn_and_Taxis_(board_game)Sons of Anarchy. Based on the TV series, each player runs an outlaw motorcycle gang where the object is to make the most money. Another “worker placement” game. As a rule, if a board game is named after a TV series or film it will probably be a poor game; Sons of Anarchy is one of the exceptions. The other exceptions to the TV series rule that I’ve played are Firefly (good with two players) and “Spartacus: A Game of Blood & Treachery” (which requires at least three and preferably more players).
Just before lockdown we played our one and only game of “Horrified”, which feels like a Hammer Horror film turned into a board game. Here a small town is under siege from various monsters such as Frankenstein, The Mummy, Dracula and Wolfman. The players work as a team to protect their small town from these monsters, each playing the sort of character that you’d expect to see in a Hammer Horror film / HP Lovecraft book. Horrified works well with two players and is pretty good as a one-player game.
Don’t play Monopoly. It’s a very poor game, particularly when you have more than two players because the elimination of players during the game means that people end up sitting around when they’d rather be gaming. As a rule if a game is well known amongst people who don’t regularly play boardgames then people who do play a lot of boardgames won’t like it. Far too many have player elimination and rely too much on luck.
I used to play a lot of Avalon Hill’s “Squad Leader", the definitive small unit wargame covering the Eastern Front of World War Two. This is a two-player game. Complicated rules, it isn’t a Eurogame. It isn't a game for beginners, though of the games of its era it's one of the easiest to learn.