Thanks for telling us about your favorite hobby, how many you have played, where it fits in your personal spectrum, that it has been around for many years, and giving us some titles!
German-style board game is a term that refers to specific game mechanics. The reference to Germany stems from the success of the "The Settlers" board game. In contrast to well known American games, like Risk or Monopoly, the game ends with all players still in the game and it introduces mechanics where players have to interact (trade) with each other to advance.
The Settlers obviously isn't the first game with those mechanics. Magic the Gathering and others might have been available a lot earlier. Still, The Settlers made these gaming principles popular to the broad masses. Interestingly, the German gaming market became huge mostly through the influence of this game as well.
> Monopoly, the game ends with all players still in the game and it introduces mechanics where players have to interact (trade) with each other to advance
This is needed in Monopoly, too. Unfortunately the meta-game is such that most people play the game the way they played when they were 9, and can't handle things like declining to purchase something in order to start a bidding war for a property two other players want. If Monopoly were freshly introduced, clever strategies of what to trade would develop. (I recently won a game with no color monopolies, just the railroads.)
I understand the ire some game fanatics feel at Monopoly, and some of it is deserved. It is using "technology" almost a century old and should do more to accelerate the end of game. But a lot of the problems with Monopoly are because people don't play by the rules, or they expect the game to suck and so it sucks.
German style board games are so named because of the influence of a few designers in Germany, and because of the popularity of some games in Europe.
Luck is reduced. Planning and skill is increased. They can be short fun easy to learn games for a wide age range (Carcassonne; Bohnanza), or they can be harder more in depth games that take longer to play (Le Havre).
Usually there's an element of competing against the game itself, not just against other players. Usually they avoid player elimination (unlike monopoly).
Ticket to Ride is a great game. You have some destination cards. You draw train cards. You use the train cards with your train counters to build routes (for points) on the destination cards. Routes consist of a bunch of shorter routes, and these are limited so other players can claim them. The luck comes in drawing good destination cards, and what train cards come up. You can try to cobble other players by claiming their routes, but you really need to claim your routes. (This isn't a great description. See Board game geek for better.)
Carcassonne is a 'simple' "draw a tile, place a tile" game. You need to build towns or farms or roads, while stopping your opponent doing the same.
Monopoly gets harsh treatment among some people. I tend to agree. If you play it properly (with all the auctions, and with the intent to drive other people out of the game) it's okay, but vicious.
There are quite a few. From the top of my head: Pandemic. Even non-European is not unheard of. For example my all-time favorite: Robo Rally (http://en.wikipedia.org/wiki/RoboRally)
Sort of cross-over between Pandemic and Robo Rally... you play and win/lose as a team like in Pandemic, but much of the mechanics is "programming" your actions like in Robo Rally.
RoboRally is one of my all-time favorite games. The only down sides are that it can take a long time to play and finding the original expansions can be difficult and expensive (sadly they haven't been completely reprinted yet).
Fun fact, the inventor of RoboRally, Richard Garfield, was asked to create a game that was more portable and he came up with Magic the Gathering.
"Simply put, Eurogames tend to focus on streamlined, well-balanced play, with a mimimal theme and more abstracted game mechanics, while Ameritrash focuses more on theme and dramatic gameplay." -- http://boardgamegeek.com/wiki/page/Ameritrash
While not really a part of definition, eurogames tend to have fewer but well thought out pieces, cards or mechanics. Often a great deal of depth comes from fairly simple rules (like Dominion or Tigris&Euphrates). In contrast, ameritrash games tend to have lots and lots of cards and dice rolls which are often poorly balanced. Examples are Arkham Horror and Bang!
Another common trait is that eurogames tend to have less interaction and feel more like a race (Dominion, Seasons) than an elimination game. It has good sides - it's harder to gang up on the leader, making him lose for arbitrary reasons. In games with high interaction kingmaker syndrome can pop up - you no longer have a chance to win, but you can determine who gets knocked down to 2nd spot. On the other hand, sometimes it feels like you can play without looking at your opponents. Some games manage to strike a balance. Cyclades has a rule which makes it almost impossible to eliminate another player, AND it allows a badly damaged player to steal a victory. This is because the first player to own 2 Metropolis wins, but they exist on the board and can be conquered, not just built.
As a german the question is rather what is not a german-style board game? I hardly know any board games not designed and produced in germany. Even the rest of the world seems to play mostly german board games. Like germany has a monopoly on board games (which is ironic as Monopoly is a popular non-german board game).
Funkenschlag (Power Grid) is a typical German board game, and great fun. It is well-balanced, and contains map-control, auctions, and resource management:
http://boardgamegeek.com/boardgame/2651/power-grid
The Settlers obviously isn't the first game with those mechanics. Magic the Gathering and others might have been available a lot earlier. Still, The Settlers made these gaming principles popular to the broad masses. Interestingly, the German gaming market became huge mostly through the influence of this game as well.