Carcassonne

I see Carcassonne as a good entry game into the whole board game business. It's simple, yet quite complex, it doesn't take too long, yet there is a good deal of strategy present. Unfortunately it took a turn similar to the Settlers of Catan and Talisman - the core game was expanded by many and many expansions and while some actually enhance the gameplay, some just crumble the core into a nonsense (I'm looking at you, Princess and Dragon).

The goal is to have the most points at the end of the game. You achieve that by expanding an area around the starting tile by adding another tiles. You can only add a tile in a fashion that it graphically connects with the already-laid tiles and doesn't clash with any neighbouring tile. The tiles have four different features on them, but not always they have all 4 of them present. There are roads, meadows, city and monastery, and connecting them to each other always somehow yields points.
Each player has a number of men to their disposal and to earn the points, the features on the tiles must be claimed by the men. You can only claim a feature on a tile that you are laying down and only if there noone else on the feature already - it sounds complicated but let me give you an example: There is a road sprawling across three tiles. You can add a tile with a road on it to either of the road's ends and claim it whole by putting your man on it. Then, when you finish the road by putting an end to both ends (the end might be a village or a city - that's where the road ends), you take the man back and earn one point for ever tile of the road.

Roads are an easy way to get money and so are the monasteries. You only have to claim the monastery and then put 8 tiles around it to earn whooping 9 points. But the battlefields of the game are cities and meadows. You can connect yourself to someone else's city (or a meadow) by starting your own unconnected city (or a meadow) and then trying to put new tiles in a way that connects your "new city" to the other player's city. Then you can fight for dominance, that is you try to have more men in that city by creating new cities with your men in it and connecting them to the other city. If all the players in a city have the same number of men in it by the time it is finished (closed), all of them earn the same amount of points. If there is a dominant player in the city, they only get all the points.

starting the game

And then there are meadows, the feature that gives probably the most points, but only at the end of the game. Meadows are the green fields everywhere and players can claim them as well. At the end of the game, a dominant player on a meadow gets a number of points for every closed city that is neighbouring with the meadow.

The strategy of the game comes from trying to get into someone's city or a meadow and blocking others' attempts to get into yours. And while you usually can decide on your turn in about a minute, the board doesn't change too much in a round, so you can prepare your turn in advance. There is a bit of luck present when you take a random tile from a pouch but that's about it. You have to think and plan. And there are some actually good expansions (like Traders & Builders) that upgrade the core gameplay by adding a few new tiles and two or so new rules.

Honestly, I recommend this game to anyone. It's as good for the new players as for those with experience in playing, though I believe that anyone who's playing board games regularly knows Carcassonne very well :)