I need to develop an extension for a Joomla website which should have this functionality:
Can someone point me where I can find the relevant information?
The building block for a Joomla extension is the so called component. It's important to understand the different extension types. If you don't know how things are called, you don't know what to search.
My advice is to first experiment with existing tutorials. Focus on understanding Joomla and not on fulfilling your specifications.
Creating a component
First step would be to build a basic component. There are numerous tutorials. Developing a MVC Component. Building Joomla Components.
Current user object
Read the basics on Accessing the current user object
Determine user permissions This is rather a complicated thing to grasp at the first time. Read on the subject by searching for Joomla ACL - Access Control List.
Providing options in the component
Short introduction is provided at How to get component parameters?
