To resign | To quit = to give up a job or position by telling your employer that you are leaving.
Have you heard of her intention to resign?
He quit the job.
To retire = to leave your job or stop working because of old age or ill health.
In 1974 he retired from the museum.
To be dismissed | To be fired | To be sacked | To lose your job = to be asked to leave a job, usually because you have done something wrong or badly, or sometimes as a way of saving the cost of employing you.
The military commander has been dismissed.
The boss fired him for neglect of his duty.
After a further four-month extension of her probation period, she was sacked.