There are two ways to achieve this, and each have their purpose.
NOTE: Server administrators can read these messages as well, so they are only private to other players.
The easy way, is used when you are near someone that you can easily see with your reticle.
Point your reticle at the person and press U
You have now initiated the private chat. Type your message and press enter. The text will be pink, indicating that it was a private message. This is only for people that are within sight. The default button is U. To change this, open the console and type /bind key messagemode3 and replace key with the key of your choice.
If you aren't near the person you want to chat with, it's a little more complicated, but is easy to remember.
Start by opening up the console (SHIFT + ~) and type:
This will give you a list of all the players in the server, with a number to the far left of each name. Note the number of the person you want to chat with.
Then you just type:
/tell # your message
Replace # with the number you noted earlier, and replace your message with your message. Then press enter and your message should send in pink.
To save time in a lengthy chat, press the up arrow in the console to restore the previous command, and just delete your previous message, so you don't have to type /tell # every time, unless it's a long message.