My understanding is that it's not so much a law as it is a contract between the union and the employer that prohibits the employer from hiring people who don't join the union.

Ah ok, but that seems very different from the comment I was replying to, which said "In some countries/industries you have to be a member of an union by the law. Just by working in industry, union is forced upon you by law, and you have to pay their fees." - that seems pretty drastic, so I was specifically curious about examples of law-mandated union membership/fee payment.