An internship is a temporary position with an emphasis on on-the-job training rather than merely employment, and it can be paid or unpaid. If you want to go into publishing, you might have to take an internship before you are qualified for an actual job.
To intern means "to pass inside something," and when you have an internship, it's like you've temporarily gotten permission to be inside a workplace. If you're just completing your medical training, then the next stage is to get a medical internship, in which you are supervised by more experienced doctors. Sometimes your internship is part of your "residency," your first hospital job.