Eurogames, or German-style games, are a broad class of board games usually contrasted with ameritrash. They're particularly popular in Germany and many of the most archetypal of the genre were first published there including Settlers of Catan, Carcassonne, and Agricola. In the last decade or two they've developed a worldwide following so the term eurogame, like ameritrash, is more a generic description than a label of a game's actual origin.
Most eurogames tend to have:
Relative Simplicity
Eurogames often have limited rulesets and can be explained in 10 minutes or less. This obviously makes them easier to teach and faster to play (especially for non-enthusiasts), but can also strike people as bland and uninteresting.
Lack of Conflict
Eurogames usually minimize conflict. Competition over resources is resolved simply by one player claiming it first. Many eurogames also incorporate trading and negotiation, which emphasizes cooperation.
No Elimination
Players are not eliminated mid-game. Scores are often kept hidden and there are frequently large bonuses awarded at the end that can launch a trailing player into contention for the win, keeping players engaged throughout.
Bounded Gameplay
Games usually end after a certain number of rounds, preventing one player from snowballing too hard.
Limited Luck
Most Eurogames have limited randomness; even roll-and-move is fairly uncommon. Additionally, players can usually make decisions after rolling (or drawing, spinning, etc.) rather than before. As an example, instead of declaring an action and then rolling to see if you were successful, you would roll first and then decide what you want to do. For many this feels more satisfying than failing an attempt.
Weaker Theming
Eurogames tend to have more generic gameplay regardless of the theme. In many cases you can drop the same mechanics into a scifi, fantasy, or historical setting and the game would be the same. It's also been noted that American games tend to focus on conflict and battle whereas eurogames more often focus on building. The thought is this reflects the postwar experiences of each--America came out of World War II with the flush of victory and an booming economy, while Europe (and Germany in particular) spent decades rebuilding from it.
Settlers of Catan is easily the best-known eurogame. Others include Carcassonne, Ticket To Ride, Power Grid, Splendor, and Kingsburg.