steve B:
Welcome to BGG. I think you'll find it an amazing website with an unbelievable amount of information on board games that will take months or years of surfing to appreciate.
BGG also boasts an incredible community. Like all communities, we have our differences that occasionally flare up. Sorry about my initial response of "uh oh" but you accidentally touched on probably the most divisive of those issues. The "Ameritrash" debates/wars got so intense a while back that a couple user were actually banned from BGG (don't worry that was an exception and it took some serious flame wars for it to happen.)
In any case, to answer your original question, 'Ameritrash' and 'Euro' are two terms used as short hand to refer to game types. (The term eurotrash is used rarely if ever here, you'll see 'Eurosnoot' more often).
The definitions are far from precise (and many games either cross boundaries into both camps or simply defy easy categorization), but most BGG users will recognize the general point you are making when you use the terms.
Ameritrash: games that revolve around a specific theme (usually fantasy or sci fi, and usually involving conflict) and where the game's mechanics are built to illustrate/recreate the 'real world' mechanics of that theme. Rules simplicity, perfect play balance, and short playing times are not necessary, if they get in the way of making the theme 'come to life.' Ameritrash games embrace some randomness and dice.
Euro: games that stress streamlined and clear mechanics. They usually have reasonable playing times, and don't stress conflict. They also don't stress random elements, least of all dice. The term "Eurosnoot" has come about due to some folks perception that fans of Euro games "look down" on other game types.
Wargames: Games that depict (usually) a historical battle, war, or campaign. Often very detailed, and complex (but not always).
There is a lot more that can be said, but I'll leave that to others.
Again, welcome to BGG. Be careful, this can become a full-blown addiction pretty quickly.