REDUCE Run mode is a subsidiary of REDUCE mode, in the sense that it requires REDUCE mode to be loaded before it is loaded (which it tries to ensure automatically). It provides an interface that allows you to run a command-line (as opposed to a GUI) version of REDUCE interactively in an Emacs window, with input from and output to that window.
In fact, it allows you to run multiple versions of REDUCE simultaneously and independently in multiple windows. By default, it runs the distributed pre-built versions of CSL and PSL REDUCE, but it can easily be configured to run other versions of REDUCE, such as ones you have built yourself from the development sources, including REDUCE on Common Lisp.
You can also run a REDUCE source code file or an Emacs buffer containing REDUCE source code (in REDUCE mode) as a complete REDUCE program in an independent window automatically named from the source code file or buffer.
REDUCE statements, procedures or general regions of code in a REDUCE mode buffer can be sent to REDUCE running in another buffer. If REDUCE is running in multiple buffers then you select which buffer to use. If you try to send REDUCE code to REDUCE but REDUCE is not running, then REDUCE Run mode will (optionally) start REDUCE automatically.