1. Refers to the act of an Understudy, Standby, or Swing learning the lines and blocking of a specific role and being able to perform that role at a moment's notice.
The Understudy and Standby both cover the lead role.
Can also refer to the act of one of these actually performing the role.
The Understudy will be covering the lead role this evening.
2. Word used to protect a control console in the theatre. These are typically called dust covers.
When the operator was done controlling the mixer, he placed a cover over top of it.