It's simple enough. American-bred Boxers (or American-line) Boxers are those that were bred in America, or whose direct ancestors are American. German-bred (or German-line) are those whose direct ancestors are German. German "style" Boxers in the US are simply the product of charlatans trying to get more money for their dogs. "Euro-Boxers" are apparently those from anywhere in Europe (except the UK), and the differences between one Euro Boxer and another can be more pronounced than the differences between a North American Boxer and a German one.
There are also several, several discussions in the archives on this topic. A simple forum search will give you more information than you ever wanted to know.