"Euro" game is a term that gets thrown around a lot without much of a definition. Frequent board gamers probably have an intuition about what the label means, though Euro games can be a bit like the "alternative rock" or "pornography" of the gaming world: you know one when you see one, but it's almost impossible to give a precise definition of them.
Before I try, a little on their history. Euro games are so-called because they originated in Germany, and sometimes you'll hear them referred to as "German-style" board games for that reason. Even today, many prominent game designers are German and board game competitions are held in Germany. But the scene is no longer strictly German, or even strictly European, so maybe "Euro-style" is a better term for it.
Whatever term you prefer, there's no one definition that encompasses every Euro game, and it's even possible for informed board gamers to disagree on whether a particular game is Euro-style or not. Common examples include Settlers of Catan, Ticket to Ride (both considered "gateway" Euro games), Carcassonne, Puerto Rico, and Agricola; virtually everyone agrees that these are Euro games, but what do they all have in common? Here are some representative traits, roughly arranged from most to least characteristic of a Euro game.
Deliberate balance of strategy and chance. Perhaps more than anything else, a careful combination of skill and luck is a hallmark of a Euro game. So-called "American-style" board games often require relatively less skill intensity and have fewer optimization decisions to make than Euro games; "roll the dice" and "draw a card and do what it says" are mechanics familiar to most gamers that rarely appear in Euro-style games. On the other hand, "classic strategy" games like Chess and Go are so purely skill-intensive that their outcome is often determined solely by the skill of the players. Euro games occupy a sort of middle ground that makes every game intriguing in a way that other classes of board games can't be.
No player elimination (and the occasional re-balancing to avoid it). In Monopoly, if you're bankrupt and own no property, you lose. In Risk, if you lose all your territories, you're out of the game--and who hasn't sat through the immensely frustrating two hours that follow one single bad turn in the worst position at the start of Risk? While it's possible to get so far behind that you have no practical shot at winning, virtually no Euro games contain mechanics for players to be actually removed from the same. In fact, some swing the other direction: Power Grid, for instance, is a Euro-style territory control game (with a few tenuous similarities to Risk), that gives players who control the least territory the chance to buy the cheapest resources.
Separate win conditions and end-game conditions. In both American-style board games and classic strategy games, the victory condition is strongly coupled to the end of the game. In Clue, the game ends when one player successfully accuses the murderer, and the player who made the accusation is the winner. In Chess, the game ends when one player forces the other player's king into checkmate, and the player who forced the checkmate is the winner. In contrast, Euro games tend to last a set number of turns or end when a specific condition is met that's not necessarily the victory condition. For example, a hypothetical Euro-style murder mystery game might end after someone successfully accused the murderer, but the winner would be the person who accumulated the most victory points along the way, not necessarily the person who made the accusation. A Euro-style territory control game might end after six turns, and the person who controlled the most territory then would win.
A modest level of abstraction and a narrative to describe it. Classic strategy games tend to be very minimal and abstract. There's no narrative to go along with Go. On the other hand, some war simulation games supply narratives of individual battles that the game is supposed to stick to. Euro-game narratives tend to be a lot less definite and just a little whimsical; the resource management game Last Will has the players assume the roles of nephews of a recently deceased man who are trying to outspend each other. Lacking the narrative of farming, Agricola becomes a game about colored cubes and cardboard; with it, you're raising animals and growing crops.
Moderately complex rules with very simple game pieces. If you've played a Euro game, you might have marveled at the dozen-page rulebook that invariably accompanies it. Because a lot of these games--at least at first--came from Europe, they needed a way to bring the game into another language without completely revamping the game. As a result, the pieces tend to be as simple as a set of Ikea directions, mostly pictures and numbers, and that design style has become representative of the genre.
Descriptive sub-genres. Because gamers love categorizing things, they're not just Euro games! They're "deck-building" or "territory control" or "resource management" or any number of other monikers.
If most or all of those describe the game you've been playing, you might be playing a Euro game!