Politicians and bloggers have been throwing around that term quite a bit lately. I've searched briefly for examples and definitions of what exactly is "European-style Socialism" and only find people using the term, not anything that defines the term and points to specific countries and practices in Europe.
Can anyone explain this for me?
Not me, and I think you're asking the wrong question. The very vagueness of the term "European-style Socialism" is an important part of its appeal to American conservatives. One the one hand, you have "European-style Socialists" like Lukashenko in Belarus. They torture their opposition in jails, would still run Gulags in Siberia if they were in charge of Siberia, and have the government run virtually the whole economy. On the other hand, you have "European-style Socialists" like Tony Blair in Britain. They approve of free markets, and merely supplement them with a decent safety net.
These two sorts of politicians have nothing to do with one another. But by calling them both "European-style Socialists", American conservatives can smear America's very moderate social reformers with the stench of the Gulag and the KGB. So don't get confused when you don't understand what "European-style Socialism" means. That's the point. You're not supposed
to understand it!