I am working for a company who wants me to test and cover every piece of code I have. 
My code works properly from browser. There is no error no fault.
Except my code works properly on browser and my system is responding properly do I need to do testing? Is it compulsory to do testing?
Whether it’s compulsory depends on organization you work for. If others say it is, then it is. Just check how tests are normally written in the company and follow existing examples.
(There’re a lot of ways Django-based website can be tested, different companies do it differently.)
Why write tests?

Regression testing. You checked that your code is working, does it still work now? You or someone else may change something and break your code at some point. Running test suite makes sure that what was written yesterday still works today; that the bug fixed last week wasn’t accidentally re-introduced; that things don’t regress.
Elegant code structuring. Writing tests for your code forces you to write code in certain way. For example, if you must test a long 140-line function definition, you’ll realize it’s much easier to split it into smaller units and test them separately. Often when a program is easy to test it’s an indicator that it was written well.
Understanding. Writing tests helps you understand what are the requirements for your code. Properly written tests will also help new developers understand what the code does and why. (Sometimes documentation doesn’t cover everything.)
Automated tests can test your code under many different conditions quickly, sometimes it’s not humanly possible to test everything by hand each time new feature is added.
If there’s the culture of writing tests in the organization, it’s important that everyone follows it without exceptions. Otherwise people would start slacking and skipping tests, which would cause regressions and errors later on.
is testing compulsory if it works fine on realtime on browser