Is that a specific political party, or just a term thrown around?

Usually what people in capitalist countries actually mean is social democracy, also known as the “Nordic model”: capitalism with a social safety net.

Democratic socialism means socialism of some form or another, usually one that isn’t Marxist-Leninist. I don’t think it’s a very well-thought out term, since Marxism-Leninism follows democratic centralism.
Michael Parenti: Left Anticommunism: The Unkindest Cut