How Can I Display Categories Description

From the category_description( $category_id ) parameters description:

Category ID. Will use global category ID by default.

This means that if you are intending to use the global id, you don’t need to insert the variable $category_id which may be undefined in your use of the function. I’m assuming it is undefined because it is not listed in the global variables. This is the default use of the function which will use the ID of 0:
echo category_description();

Otherwise, if you want the description of a specific category, pass the ID of that category as an argument, such as:

echo category_description( 3 ); //gets the description for category with the ID of 3