Well, the answer is both. You will have to test the functionality of that module as well as the other modules. It also depends on the module you are modifying.
All the modules should be tested because new changes might have an impact on the other modules as well. You can, therefore, differentiate by the stress given on the module which is to be tested.
I think this scenario will explain the answer to your question well.
If Module A is modified, Module B is depending on module A, and Module C is a general module independent of module A.
In this case, you will first test the module A in depth. Your next stress will be on the module B. But what about module C? This module will be tested as well but with less stress, because module C does not depend on module A for its functioning, rather it depends on module B.
If you are a white box tester, you must be knowing which modules are to be tested and which ones can be affected. But in case you are a black box tester, you will be required to do Regression Testing as well.
Regression tests should be carried out on only those modules associated with the modified module.