Our shul library has been neglected for many, many years, and b'h' we are finally cleaning and organizing it. We have some great books, but I'd like to know of some resources identifying important books that we should make sure we have in our library.
Libraries vary significantly from shul to shul based on the congregation but there are some standards that are common across the board and which would be missed if anyone tried to use the shul library for studying.
Siddurim. These aren't really part of the library, but a basic necessity because most of your congregation will want something to pray out of. It's worthwhile to have at least some in translation and an assortment of nuschaot different from the one the shul uses so guests will feel comfortable.
Chumashim. Also a basic necessity in any shul. Translations and rashi are helpful.
Mikraot Gedolot on all of Tanach. These come with many exhaustive commentaries and are meant for in-depth study, not for following along during Torah reading.
Talmud bavli. The second most important Jewish text. A functional shul library should have at least two sets because people tend to learn gemara in pairs. Since you mention that you are "cleaning out" your shul library, please allow me to beg you on behalf of gemara learners everywhere: please, please get a new shas in addition to your old ones. The newer editions are much more legible, and contain more and better-edited supplementary stuff.
Two sets of Kehati Mishneh
A good edition of the Mishneh Torah.
At least one set each of Mishneh Berurah and Orech Hashulchan.
Most shul libraries will have many more books than this, depending on the interests of the members but these are the basics. All of them are available in English and some in other languages as well. Which ones you get, and which editions obviously depends on the needs of your community.
The Lubavitcher Rebbe emphasized having at least the following Seforim in order to fulfill his directive, known as "Mivtza Bayis Molei Seforim", of filling up Jewish houses with Seforim (list in no particular order):
Chumash, Tehillim, Siddur, Tanach, Kitzur Shulchan Aruch, Tanya, Talmud. (See here)
A Tzivos Hashem Children's handbook (not available online) also lists a Haggada.
Of course this is only a small list, and should be expanded upon according to the means and needs of the specific shul.