Something that's treasured is cherished or prized—it's very important. If you're a baseball fan, your most treasured possession might be a baseball cap signed by Jackie Robinson.
The adjective treasured comes from the noun treasure, and that's exactly what treasured objects and people are, things that we treasure or value highly. So while some of us might have treasured valuables (silver or jewels, for example), for others, our most treasured things may be our favorite books, a musical instrument, or a pet guinea pig.