Eurogame is a collective term for games that tend to be more abstract in theme and more family oriented. These games are typically not sold in the retail mass-market stores, but through specialty shops and online. Many were developed in Europe (mainly in Germany), hence the term "eurogame".
Many of these games have a fixed ending point and usually take 1-2 hours. Some also include mechanisms to keep you involved in the game when it is not your turn (such as trading, negotiating, etc) - the result is more involvement and less down-time during play. Luck is a much lower factor in many of these types of games, but still plays a small role.
Some examples of eurogames are Settlers of Catan, Ticket to Ride and Carcassonne.
Wikipedia has much more info.
BoardGameGeek.com is the internet hub of all things board game. BoardGamesWithScott.com has video reviews of board games and is an excellent place to start. Check his New to Boardgames for the Boardgaming 101 episode for a good intro.
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