Clojure is a language compiler, implemented as a .JAR on the JVM, so I'm not quite sure what they're getting at. ClojureScript is an equivalent, targeted at JS in the browser, so I'd say they are the same language but with different libraries for the 2 environments. Leiningen is the REPL and configuration tools for an interactive environment.
The interesting bit is CIDER which is a client-server pair, the client part living inside emacs, written in eLisp and the other living in nREPL, part of Leiningen, written in Clojure.
I think I've understood what @paulspencerwilliams was doing in his demo to the Functional Programming Meetup, for the first time; connecting from emacs windows to 2 different instances of nREPL, dynamically hacking the Clojure back-end code and ClojureScript front-end code at the same time.
I'm still quite impressed that you can hack the emacs session that you're running to do the edit so you can add a function you need. See: recursion. No wonder Lispers always have crazy hair.