Grossly overgeneralizing here, the difference between a "German-style game" or "Eurogame" and what is affectionately known as "Ameritrash" among boardgame enthusiasts (though there are games in both categories designed in both the US and Europe) is that Eurogames tend to be games involving strategic optimization to earn points, whereas Ameritrash tends to emphasis direct conflict between players. Though it's more of a spectrum than a dichotomy.
The big three light (as in easy-to-learn) Eurogames that really went mainstream are Settlers of Catan, Carcassonne, and Ticket to Ride. Of those three, only SoC really offers the possibility to be a dick to another player, and even there not to the degree you would see in direct-conflict games.
Note, among other things, that it is rare to have a runaway leader in these games, and in the cases of Carcassone and TTR, it's almost impossible to tell for sure who is winning until the game is over. That tends to keep everyone engaged and enjoying themselves.
If you want examples of heavier, pure Eurogames, take a look at something like Puerto Rico, Argricola, or Power Grid.
Watch Settters of Catan being played
Pfft. The only interesting way to play Ticket to Ride is to strategically block the other players' routes.
Rules: