I would add a note of mild disagreement with a couple of the prior posters on unions protecting bad nurses. I'm chief nurse rep - what many unions would call a shop steward - at my hospital in California. In effect that means I deal with a lot of the disciplinary issues from the union side. I don't believe it is all that hard to fire a nurse who genuinely needs to be fired. There is a requirement for due process and documentation, but due process is a pretty basic American value. A manager needs to document how the nurse is failing to meet expectations, give them clear notice of that, give them a reasonable amount of time to bring their performance up to par and if they are still unable to meet reasonable standards they can be terminated - assuming that their failure rises to the level of something a reasonable person would terminate them for.
Of course if there is behavior that is really dangerous to patients, all the notice and time requirements go out the window.
I'm convinced that what happens a lot is that managers just don't want to go to the trouble of doing a termination properly, so they blame "the union" for their own failure.
When a termination seems to be genuinely unfair, the ultimate recourse in a union environment is to take it to arbitration. This is not done lightly, since arbs are expensive and most unions have mechanisms in place to prevent local reps from going to arbitration frivolously. When you do go to arbitration, an arbitrator looks at the question of "just cause" for a termination. The just cause standard rquires that the arbitrator look at issues like:
Was the policy violated a reasonable policy, did the employee know about the policy, were they given an opportunity to correct their behavior, balancing the severity of the violation against the employees past record, etc. That last one means, for example that you might have two employees who committed the same offense. If one of them is a fairly new employee with a couple of other issues on their record, firing might be reasonable. but perhaps firing would not be considered reasonable for the same offense if there is a 20 year employee with an otherwise excellent record.
My experience is that many American workers are under the false impression that they have some sort of legal right to basic fairness in their workplace. In the absence of a union, that is mostly not true. There is a short list of reasons for which you can't be fired - you can't be fired because of your age or gender or race, for example, (assuming you can find a lawyer to take your case and can prove that was the reason) but in most places, in the absence of a union you can be fired for no reason at all.
Having said all that, I don't think that's the most important reason for unionizing by any means. Most nurses won't get in disciplinary trouble and most disciplinary actions are justified. Union workplaces generally do better on wages and a lot better on benefits than non-union. And the better nurses unions also provide an effective voice for nurses to protect the quality of practice both in the workplace and in the state legislatures - where all the laws that affect our practice our made. Examples of unions that do a pretty good job in those areas might be NYSNA, Massachusetts Nurses Assn., PASNAP in Pennsylvania and CNA/NNOC. In terms of professional voice, a lot of nurses are also represented by unions that are not mainly nurses' unions. Everyone from Steelworkers, Laborers, Teamsters, Teachers unions, etc represent a few nurses, but in my opinion they do much less well on the professional side, may or may not do OK on the bargaining side.