These are used to send role-related Night Choices in to the moderator. They are also commonly used by the moderator themself – if one player needs to be told private information (such as Cop results or what their own role is), but the information should not be public, the moderator will typically send the information via private message. On some sites, players who are specifically told they can speak to each other can use PMs for that purpose as well; however, the usual method nowadays is to set up a private topic for such players instead.
Because many players will be notified via email when they receive a private message, the private message system is also widely used to send prods; if a player is not paying attention to the game, a private message can often be the only way for a moderator to send a reminder in a way that that player will notice.
Players are in general not allowed to PM each other unless otherwise told they can.