I've heard a lot about ZeroMQ and its advantages, but I'm not really sure what it is. What are some example uses, what is it trying to replace (if anything), what problem does it solve, what are the alternatives out there, etc.? And, what is a "messaging library"?
ZeroMQ, as it's name suggests most probably is a messaging provider. A messaging API is needed to send and recieve messages using these message providers. And you need to integrate these providers with your application server (view documentation). Some MQ supports multiple platforms like Ruby, Java, Php and Others. It is used for loose coupling between two modules in an enterprise application. If you are a Java Programmer, refer to JMS Specifications (Java Mesaging Service) at Oracle's site.
What is ZeroMQ?