To private message someone click on the player, then click on the name tag. A menu will appear, select Private message. Or open the Social tab (O), click on a friends name and select Private message.
PM playermenu
PM friendmenu

A window will appear where you can type a message to the other player.

Note that when the player closes their private Message window the other player isn't notified of this. So the player may wish to sign off their private message to notify the other player the conversation is finished.

PM window
Clicking the minimise button PM minimisebut will minimise the Private Message window to a small icon located in the bottom right of the game screen.
PM minimise

When the player recieves a new Private Message the icon will flash.

PM minimise new