Often in this situation, people have one or more bad reactions: They stew about silently, or they just disregard the boss's instructions.
Neither of these is a good option. If you disagree with your boss, you should offer up your own viewpoint. (Disclaimer: This assumes your boss is sane and reasonable, not a tyrant or a lunatic. If she's one of those, that's a different topic.) Often, workplace disagreements arise when two people have different pieces of information about something. It's possible that you know something your boss doesn't know. Figure out what that might be, tell her, and see if that changes anything. At the same time, be open to new information she might give you that might change your own viewpoint.
However, if she still disagrees and you feel very strongly about it, you should talk explicitly about the fact that this is an area of disagreement. As a manager, I would much rather know if an employee vehemently disagreed with me about something. I'd want the opportunity to hear her perspective and consider her viewpoint, and to know how strongly she felt, even if I didn't ultimately come around to her way of thinking. After all, a good boss has hired you because you're smart; think of it as fulfilling your professional duties to be honest about your assessment.
You want to do this in a polite and collaborative manner, of course--I'm not advising being a jerk about it. And do it in a way that demonstrates that you understand that in the end your boss is the one who will need to make the call.
If her final decision is one you still disagree with, you can say things like, "I really feel strongly about this. Would you be willing to allow me to try it my way and we can see how it goes?" Of course, if she refuses, you'll need to accept that, since that's the nature of having a boss. You also can't keep the debate going forever. More than two separate conversations is usually overkill, depending on the specifics.
But the whole point is to be open and candid. Figure out your differences, see if they can be resolved, and if they can't, decide if you can live with that.
Alison Green is chief of staff for a medium-sized nonprofit where she oversees day-to-day management of the staff as well as hiring, firing, and staff development. She is working with the Management Center to coauthor a book on nonprofit management. Her writings have been published in the Washington Post, the New York Times, Maxim, and dozens of other newspapers. She blogs at Ask a Manager.