When a controller instructs you to "verify," he or she simply wants you to confirm information. For example, if a controller asks you to "verify assigned altitude," he is asking you to tell him your current assigned altitude.
If you ever hear this phrase or one similar to it, take a quick look at your instruments and don't be surprised if you aren't where you're supposed to be. Many controllers use the request to verify as a friendly reminder that you have drifted from your assigned course or altitude.
Sooner or later you will hear a controller tell you to "maintain visual separation." For example, you are in the pattern at a busy towered airport. The controller tells you that you are number two for landing behind a Piper Archer and instructs you to maintain visual separation. That means you must see the other airplane and maneuver as necessary to keep a safe distance from it.
Another time you may be asked to maintain visual separation is when you are receiving flight following. If the controller has identified traffic and you have responded that you have it in sight, you may be asked to maintain visual separation by watching, and staying away from, the other aircraft until it is no longer a factor.
A controller may also do the separating. When a tower controller has two aircraft in sight, he may maintain visual separation by issuing instructions as necessary to keep the aircraft apart.