Yes, a nonprofit organization can hire employees and pay staff members. Most nonprofits will need at least a small number of employees in order to fulfill their mission. Many will also rely on volunteer support as well. For instance, a church might hire a part-time administrator, but might also have volunteers who clean, serve refreshments, offer pastoral support, and so on.
It’s important to note that board members are typically volunteers, who may receive expenses (such as for travel) but who will not normally receive compensation.