I'd encountered the term but never really given it much meaning - digging around, there's a nice, generous BGG post which suggests that where the core priority of a German game is "elegance" and the core of a wargame is "realism", the core priority of an Ameritrash boardgame is "drama".
Yeah, basically it boils down to this: in a "eurogame" (games like Settlers of Catan and Carcassonne are classic eurogames), the designer first comes up with a great mechanic, and then tries to bolt a theme to it. In ameritrash, you first get the theme, and then try to get mechanics that work with that theme. The result is that in eurogames, the theme is sometimes felt to be "pasted on", while in ameritrash, them mechanics are sometimes filled with unnecessary exceptions and other oddities to fit the theme.
In contrast to the majority, I prefer to consider these terms in relation to historical design currents. So "Ameritrash" is simply the style of design that was developed through the late '70s, '80s and '90s for the geek boardgaming market in the USA, while "eurogames" are the style of game that developed especially during the '90s in the German and German-influenced Middle-European market area when the quality of family boardgame design shot up. Both have traditionally been, just like wargames, something of their own scenes with designers taking influence from each other and striving to sell to the same target audience.
The analytical efforts to define these styles of design come on top of this factual historical matter. It's perhaps not too far from the truth to say that eurogames tend to be more cleanly designed and boardgamey, while Ameritrash games have lots of detail, theme and in general more focus on the joys of simulation than joys of a standardized game experience. But these are tendencies, and both traditions have achieved similar things in some ways. Something of a matter of taste when you pick the features you want to emphasize as the fundamental differences. For instance, I myself tend to consider simple discretionary resource bidding multiplayer games a central Ameritrash feature - these games tend to have much freedom within a player turn to hit other players, which in turn makes all of these games behave the same in regards to hit-the-leader, kingmaking and other follow-up behaviours. So when you give me a game where player vs. player struggle is paramount and the most important choice you'll make in the game is to prevent somebody else from winning, that's a red flag that screams Ameritrash for me.
Of course, today these two currents of design are more and more intermingled as eurogames have hit the American market in a big way while the hardcore eurogame audience in Europe has developed into what is essentially a geek subculture willing to dedicate serious time and money to these games, which in turn makes the hardcore eurogame resemble American games a bit more. Especially American designers have taken considerable influence from the euro side (while it can be argued that euro designers have been more aware of the American design ever since the early '90s) during this decade, which shows clearly when you look at the sort of games Fantasy Flight Games puts out, for example.
If you want an informative look at the differences between eurogames and Ameritrash, it might be interesting to look into some similar games from either side of the pond. For instance:
Both are modern hybrid designs to be sure, but the origins are very clear: Wallenstein is long by euro standards and has a traditional American set-up in that all players are striving to hit each other for victory points. However, the game is designed to make the logistics of troop movement limited so as to drastically limit the choices players have in hitting each other; it is difficult to say for sure who is winning because the game has an elaborate victory point scheme; individual parts of the game are built from classical eurogame parts, such as area control and role assignment. The end result is a game that doesn't play like a war game even when that is exactly what it is - you rarely have an opportunity to hit the leader or act as a kingmaker.
Compare to A Game of Thrones, which is an American design, a mid-length war game. It has very euro mechanics in many parts of the game, such as literal bidding, streamlined troop logistics, game end in set number of turns and discretionary resources in excess of the troops on the board in the form of card hands. However, the heart of the game is solidly American, with the players having much freedom to choose their targets, simple victory point schemes that allow players to pinpoint the leader early, simultaneous movement that allows backstabbing games and so on. While both GoT and Wallenstein are war games in theme and basic mechanics, the concerns of the players are very different in the two due to the different design cultures they ultimately spring from.