The Euros tend, as Alex said, to be folk building an 'engine' and to optimize that engine to be successful in scoring. The reference/interaction with others tends to be limited to maybe buying from a similar market or a bit of modest competition for key board spaces. Most of the time, there is not directly antagonistic play (indirectly sometimes, but that's not a major focus).
In many American style games, there are direct engagements of other players (by cards, actions, clashes, and other measures) and there is a definite expectation that direct conflict will be a significant part of the game (in order to win, you have to manage the clashes - or dominate them - while doing some other things as well, but the antagonistic mechanics are kind of core to the game (such as rents in Monopoly... but it is far more evident in Risk-style games).
One of my favourite American games is Cults Across America (different Cults sparring over who will be the most important at the end times...). Another is Monsters Menace America. A third that fits that model to me is Nexus Ops. Others that immediately come to mind are Ebola Monkey Hunt ("An infected monkey is running towards you while another scientist chases him... whom do you shoot with your dart gun?"), Sucking Vacuum (ISS going south, only one or maybe two life pods... more players than that....), and there's one about a steampunk gnomish sub is going south called Red November.
Euros are good for people who don't enjoy antagonistic aspects of games.
Now, those are only broad types. There are some that are more, or less, close to those sorts of mechanics.
Myself, I'm a big fan of 'players vs. the game' co-operative games like Pandemic and Gloomhaven (although it's a long legacy campaign game rather than just a one off).
Also for our resident submariner:
Captain Sonar is vicious - two teams run subs in a fight and they can hear each others communications and try to plot out where the other guy is and where they are and so on. It's quite tense. Submarine Escape is just kind of dorky fun. I do love the dark nature of Sucking Vacuum though.