Interrogatories are a discovery device. During the discovery phase of a civil litigation the parties share and acquire evidence regarding the case.
Interrogatories are written inquiries one party sends to the other party. Often, but not always, interrogatories are in the form of a question. Interrogatories might ask a party to describe an accident or to name witnesses. The party receiving the interrogatory must usually respond to the interrogatories in writing.
Please keep in mind that rules for interrogatories can vary from state to state and also between federal court and state court. Interrogatories in the federal courts are governed by Rule 33.
The federal court for the Southern District of New York restricts the use of interrogatories. Initial interrogatories are limited to asking for the names of witnesses and regarding the computation of damages. See SDNY Local Rule 33.3