Eurogames are board games that are actually good.
Basically, you've got five types of board and card games
1. Classic. Your chess, poker, etc. Scrabble has kind of an honorary membership here.
2. American "family" games. Monopoly, etc. Tend to be kid stuff. High emphasis on "roll and move" game mechanics. Best of the lot is probably Clue. Worst is stuff like Chutes and Ladders, which is an entirely deterministic exercise in luck.
3. American trivia games. They fill shelf after shelf in places like large grocery stores.
Most people only know about those three categories.
4. Ameritrash games. Don't get mad, this name has been embraced. These are gamer board games. Usually involve tons and tons of plastic figures, lots of die rolling, and a theme about war, something geeky like elves, or a war with elves in it.
5. Eurogames. Typically nonviolent themes, playtime varies but is usually shorter than Ameritrash games (not always), tend to be games of strategy. "Roll and move" is almost nonexistent. Sometimes themes are counter intuitive, or feel "pasted on," meaning that they don't really match the underlying mechanics of the game. Telling a Eurogamer that the game Samurai doesn't actually have anything to do with samurais will net you a look of confusion- was it supposed to? And in spite of the theme often being "pasted on," the production values will typically be high and you will typically get a lot of lush, theme related art work. This is because the primary audience for these games is adults with disposable income. A Eurogame isn't necessarily from Europe, this denotes a style rather than a place of origin.