$ibforums->member['mgroup']
$ibforums is an instance of an object. That object contains a property called "member" which is an array. That array has an element identified by the index "mbgroup".
Does that help?
class module extends module_loader is further OOP-speak. A class called "module_loader" already exists and a new class called "module" is now being defined, which will have all the same properties and methods as "module_loader" but which may well have addiotnal properties and methods as well which are about to be defined.
OOP stands for Object-Oriented Programming, and you may well need to read up on this before you will be able to make proper sense of the code you are currently reading.