Internships can be paid or unpaid but are of immense value either way. Employers like to see that a potential candidate can balance the demands of school and an internship or job. Plus, an internship gives you experience to add to your resume as well as time toward total work experience. Yes, you should absolutely include that time when asked how many years of work experience you have under your belt. Once you're several years out of school it may not make sense to do so but as a recent graduate, you should include it.
Lesson Learned: Interns don't just get coffee. One of my best work experiences was an internship at an educational exchange non-profit. Despite being an intern, I was treated as a team member and was given tasks of program officers.
Lesson Learned: Internships can turn into a full-time job. Treat the internship as you would a job. Do your best work and form connections with your colleagues. I got several unexpected phone calls before graduation asking if I'd be willing to take a newly-created full-time position there.
Have you done an internship? If so, what do you think are the benefits?