To my knowledge, there isn't a single definition of a union. In America, it seems to mean something wildly different than what I'm used to in my part of Europe.

To me, it is simply an organization by workers to collectively bargain with capital (who have their own equivalent to unions). The unions also typically offer unemployment insurance as well as income replacement in case of strikes, which are a possible measure that can be organized by the unions to force capital to come to the negotiation table. An employer has no right to know if you're in a union or not, so it would be quite impractical to try to hire only non-union members, and most companies don't mind either way, because the collective bargain agreements go both ways and can make it simpler than negotiating individual agreements with each employee.

In the US, it seems like unions are what I would call a guild - very highly coupled to a particular profession. Where I'm from they typically accept anyone who works in a general sector (say medical, government, transportation, etc.), as typically most workers in the same sector would have similar challenges and goals.