Doing something reverently means doing it in a solemn, respectful way. A restless crowd waiting to see the Pope might sit reverently when he finally arrives.
The whole congregation prays reverently together when a minister leads it in prayer. In fact, a church, mosque, or temple is the perfect place to act reverently, but you probably do it in many other places as well. Some fans stand reverently during the national anthem before a baseball game. And if your favorite musician is playing a concert, you might listen reverently, shushing anyone nearby who's talking. The Latin root is revereri, "stand in awe of."