To check your mail, click on the Mail icon on the top left corner of the gamescreen. That will take you to your main Mail screen, showing your "Pigeon Post" capacity, your mail, the sender, and the trash.
To Send a Letter: Click on yourself to bring up your menu. Then go to "Social" section and choose from either "Social Network" or "Inner Circle". Choose the person you wish to send a letter to. By clicking on them, the screen shows information on the person. On the bottom right hand will be an envelope icon that will let you send a letter.
Reading a Mail: Simply click on the envelope image and you are able to read it. From here, you can reply back to the sender or trash the letter.
Replying to Sender: When you are reading a letter, there will be a "Reply" button on the bottom right hand of the letter if the sender was from a friend. If the letter was from a Computer Player, a Reply button would not be included.