What does "The APR based Apache Tomcat Native library was not found" mean?

It means exactly what it says: "The APR based Apache Tomcat Native library which allows optimal performance in production environments was not found on the java.library.path"

The library referred to is bundled into an OS specific dll (tcnative-1.dll) loaded via JNI. It allows tomcat to use OS functionalities not provided in the Java Runtime (such as sendfile, epoll, OpenSSL, system status, etc.). Tomcat will run just fine without it, but for some use cases, it will be faster with the native libraries.

If you really want it, download the tcnative-1.dll (or libtcnative.so for Linux) and put it in the bin folder, and add a system property to the launch configuration of the tomcat server in eclipse.

 -Djava.library.path=c:\dev\tomcat\bin

Unless you're running a production server, don't worry about this message. This is a library which is used to improve performance (on production systems). From Apache Portable Runtime (APR) based Native library for Tomcat:

Tomcat can use the Apache Portable Runtime to provide superior scalability, performance, and better integration with native server technologies. The Apache Portable Runtime is a highly portable library that is at the heart of Apache HTTP Server 2.x. APR has many uses, including access to advanced IO functionality (such as sendfile, epoll and OpenSSL), OS level functionality (random number generation, system status, etc), and native process handling (shared memory, NT pipes and Unix sockets).