An "internship" is a short-term job that a student (or sometimes a young person who has already graduated from school) takes, in order to get experience in a field. Sometimes the student will get paid for their work. If so, you can call it a "paid internship". But a lot of internships are unpaid. Students do internships because it will make it easier for them to get a job in that field later if they already have work experience.
Use the word "get" to talk about being accepted for an internship or being offered an internship:
When I was a junior in college, I got an internship at HBO.