Can a nonprofit show a movie without charge at a library and not break copyright laws?

From what you say, no you cannot do that without permission. The copyright laws prohibit the public display of a copyrighted work without authorization. Displaying a work means showing a copy of the work, directly or via some device or process. A display is considered public in one of four situations: (i) when it is at a place open to the public; (ii) when it is at a place with a group of people larger than a gathering of family or the normal circle of friends; (iii) when it is transmitted to a place open to the public or a group of people larger than a gather of family or the normal circle of friends; or (iv) where it is transmitted to the public (i.e., television and radio broadcasts). In fact, under 17 USC Sec. 110, TV broadcasts and movie showings can only be displayed so long as "no such audiovisual device has a diagonal screen size greater than 55 inches, and any audio portion of the performance or display is communicated by means of a total of not more than 6 loudspeakers."