In general, apply for clerkships by submitting specific documents directly to the Judge/Justice (or other individual listed in a job posting). Apply through regular mail unless the judges is participating in the Online System for Clerkship Application and Review (only some federal judges) or requests otherwise. Note that some state courts hire through a more centralized process and require submission to the court as a whole.
Most judges and courts require: cover letter, resume, law school transcript, writing sample, 2 or 3 letters of recommendation. Some judges and courts also require items such as: undergraduate transcript, LSAT score.
The judge or court interviews selected candidates. If you are selected for an interview, you will be contacted by the court or judge by letter, phone, or e-mail. There is usually only one interview phase and the judge or court reaches a hiring decision shortly thereafter (or stops the process when an ideal candidate is found).
Once an offer is given, the successful candidate typically has to respond within 24 hours, or even the end of the same business day. Therefore, the golden rule of clerkship applications is to only apply to judges or courts with whom you would immediately accept an offer.
Many judges or courts list date ranges for accepting applications (i.e., September 7, 2017 until October 31, 2017). Apply as early in the date range as possible because some judges stop accepting applications if they find numerous qualified candidates.