So you're looking at this from the Admin user's POV.
The user is looking at an Instructor Edit page.
You have a multiple select field (box).
You need to add a new item to the select field box. So you need a separate form/lone input box with a button next to it that says "Add Category".
Server takes in Category Name, spits out Category ID and Name again.
AJAX handler processes return, appends element to the list.
1: Forming the AJAX request
2: The PHP to receive the AJAX request and return the reponse
3: The line of jQuery needed to add the category to the box
Show us what you've done, and we can try and point you in the right direction.