What do you mean by "verified stay"? Who verifies, what is that process?

A Verified Stay is a review based on a booking through Rover. That means the client PAID Rover for whatever service the client chose, completed stay, and after completion the client can leave a review. This could be a positive review or a negative review. It depends on the client's experience with the selected sitter.

Other reviews, that are not "verified" were left by friends, family or clients who did NOT use Rover and didn't pay Rover.

It just means the review was by someone who booked a service or stay through Rover. The "unverified" type are the testimonials a sitter solicits from friends, family, or people he/she may have provided dog services outside of Rover.

Its basically a way for Rover to say that someone who has an account on this site booked and completed a stay with a sitter. I'm not sure if its 'verified' with a review or if its simply just a completed (uncontested) booking.

Its a way for Rover to insure that the reviews and the process is not faked by bogus reviews and stuff like that.