Republishing a book that is out of print can only happen if either of two conditions are true. You must be the author who owns the copyright or the book has to be in the public domain. If you sold the copyrights to a publisher who has allowed the book to go out of print, you must re-acquire the rights to the book before you can either pitch it to a new publisher or self-publish it.
Realize that the copyrights for books that were published after 1977 do not expire until 70 years after the author's death. However, even if you are the author of a book that has gone out of print, you may not have the rights to republish it unless your original contract defined what constitutes "out of print" and specified that at that point, the rights revert to you. In the event that your publisher went out of business, be prepared to buy the remaining inventory and the rights. It may be helpful for you to consult a lawyer in this situation.
Prepare a book proposal to send to prospective publishers. This document includes a summary of the book, how it differs from other similar books and your author credentials. You should include information about the past sales of the book and your ideas about how to continue to market it. Discuss the reasons that the book went out of print and why you feel that it still has enough of a potential market to warrant it being published by another house. Pitch that because your book has previously been published, and therefore type set, it will be more cost effective to produce than a totally new manuscript.
Use a directory of publishers like the current edition of Writer's Market to research prospective publishers who may be interested in your out-of-print book. Decide on which publisher to approach first and then follow the publisher's directions. Some publishers prefer to get an initial query by mail or email. Others want to see the whole book first. Of course, if you already have a relationship with an editor at one of the publishers, use the relationship to get your foot in the proverbial door.
Expect to receive a smaller advance from the new publisher than from your original one. Make sure that this contract keeps the copyright in your name and that if the book goes out of print again that the rights revert to you. Of course, you can decide to self-publish your out-of-print book and then you do not need to wait until another publisher accepts the book.