The APR connector is a native JNI based connector that uses the Apache Portable Runtime libaray. Essentially is takes a parts of the code that Apache Httpd is written on and allows Tomcat to leverage the same speed and scalability as an Apache Httpd server. Its very high performant.
The NIO connector is a connector that does not block with threads. When clients connect, they are not siitingin in a blocked thread waiting for the Tomcat server to answer. Blocked threads eat up resources. With NIO, the connection gets parked, and the thread can be released to do other things. This, along with useing the CometProcessor servlet can allow an application to scale much more concurrent connections than with a standard connector.