A Direct message is a personal message sent via Twitter to one of your followers. DMs are different than mentions and @replies.
In turn, people you follow can send you a direct message.
You cannot send a direct message to a user who is not following you. You can only send a direct message to your followers.
How to send a Direct message via the web:
Log in to your Twitter account.
Click on the person icon and select Direct messages from the drop down menu.
You'll see a pop up showing your Direct message history. Click the New message button, highlighted below.
In the address box, type the name or username of the person you wish to send a message to.
Enter your message and click Send message.