Solemnness is a quality or state of being very serious and formal. The solemnness in your teacher's eyes as she enters the classroom might make your fellow students stop chattering and pay close attention.
You'll notice solemnness at any very dignified event, like a formal wedding or a serious academic lecture. There is also a sense of solemnness when the occasion is sad or somber: funerals and memorial services are often imbued with solemnness. The noun solemnness comes from the adjective solemn, which was originally used to mean "performed with religious ceremony," or simply "sacred." The Latin root, sollemnis means "annual, formal, or ceremonial."