The civil calendar is the schedule of upcoming trials for a court. If a lawyer has a trial scheduled, then he or she will get a notice from the court when that trial is near. When this happens, the lawyer is said to receive a "calling" from the civil calendar.
The civil calendar exists because there are many trials that must be scheduled for court dates. If there were only a few trials every month, then the court could just hear them as they came up. However, a single court can be inundated with trials and other legal proceedings. So, trials often have to be scheduled far into the future. The civil calendar is a way for lawyers and courts to keep track of the dates that trials are scheduled.