What is a “DM”?
DM stands for “Direct Message.” Direct messages are like tweets, but they are directed privately to one other user. You can only DM users that are following you. Someone may say “DM me for the details” - indicating that they want you to contact them privately. There is still the limit of our hundred and 140 characters on a direct message, so you may still have to ask them for their e-mail me to have more meaningful correspondence.
You can send a direct message from the web via the "message" link on the profile page.
You can also send direct messages from the status update box using the direct message command: "d + username + message." To reply to a regular Twitter update with a direct message from your home page, click the arrow/swoosh at the end of an update. This sets an @reply in the update box. Remove the '@' sign and replace it with a "d" and a space, and type in the rest of your message. If the recipient has their direct message text and email notifications turned ON, they will receive direct messages on their phones and/or via email.
Direct messages are like tweets, not e-mail
The author of the Direct Message, can delete it. Deleted messages disappear from the sender's sent tab and the recipient's inbox. There is a number next to your direct messages tab that indicates how many unread messages you have.
Be careful if you're sending a direct message from your phone. If your provider lets you type in more than 140 characters, twitter will split for you, but it will not know that the second half is also a direct message and it will be published publicly.