An "internship" is typically an unpaid position, but depending upon the employer the term is often used interchangeably with "externship." Many courts, government agencies, public interest entities, and other non-profit employers offer internships to 1L and 2L students.
Each spring, the law school coordinates a state-court judicial intern/extern program. State-court judges from around the state, in many different types of courts, participate. Most of the judges will allow students to claim class credit for the experience as an externship.
Many federal judges offer internships for 1L and 2L students. (This is not the same as a federal judicial clerkship, which is a paid, one or two-year position with a federal district or appellate judge following graduation.) The CDO will notify you of any advertised internship opportunities. Otherwise, the best way to obtain a federal judicial internship is to contact the office of the court where you wish to apply. The best time to apply for these internships is mid-December through late February. Most judges will hire summer interns by March or April.
The United States Department of Justice offers a number of paid and unpaid opportunities for law students. Please see the information contained in this DOJ webpage.
An additional resource for government opportunities is the subscribers' version of The University of Arizona College of Law's Government Honors & Internship Handbook. Baylor Law School is a subscriber to this resource. Contact the CDO for login information.