In Xenforo, Private Messages are referred to as Conversations.

You can start a new conversation with somebody via your Account page

From the main section of the View Conversations page

or by clicking on a user to show their account overview, and selecting Start a Conversation

When starting a new conversation with someone, you can type their name in the Participants box, and this will also display the usernames based on what you are typing. Select the person. You can also add multiple Participants by separating them with a comma.

Multiple Participants:

You then need to give the conversation a title, and type your message to them.

When done, click the start a conversation button.
You will then be taken to the conversation page. This will show who the participants are on the right hand side.
As you will probably notice, this looks just like a normal thread on the forum. This is basically what a conversation is, it's a private thread between yourself and 1 or more participants.

At the same time, the recipiant of the conversation will be notified of this via the Alerts system.

When they click to view the conversation, they will be shown the same window as you saw earlier.

Making a reply is the same as posting a reply to a normal thread.

The conversation window is threaded, unlike the old software. All new Private Conversations will show threaded in the main window, but the old PM's converted across will only show a single reply for each one. I've highlighted the old imported PM's in the below image.