Generally, paid interns with set hours should be treated the same as regular employees in terms of payroll. Paid interns should file a W4, and the payroll company should deduct state and federal taxes from their paycheck, just as they would for regular employees. Interns should be paid on the regular payroll cycle, as defined by your company's policy.
The only difference on paper between paid interns and employees is that an intern's offer letter should state the short-term nature of their role, in addition to any notes about benefits.
On rare occasions, paid interns can be classified as independent contractors. This is possible if their duties fit the definition of an independent contractor, meaning the intern is an individual who doesn't require supervision for their job, and provides their own tools for work.
Interns don't require a special payroll classification, and generally it's easiest to file them as regular employees.