Direct Messaging somebody on Twitter is a way to take a public Tweet, or Twitter conversation a little more private. It's the virtual equivalent of saying, step into my office, will you? It's also referred to as the Twitter back channel.
One rule: You can only DM users who have followed you and vice versa. This becomes a problem for companies who get complaints and try to move the conversation into a DM to avoid a public fued.
So, to avoid any embarrassment, before you ask somebody to DM you with more info, be certain you've actually followed them, otherwise they won't be able to.
The easiest way to DM somebody is to type D, plus a space, in front of their @username, then type your message. Or click on the little black head and shoulders icon from their profile and choose, Send a direct message.
If, after you've sent it, you a have a fit of pique that it could possibly appear on the public timeline, then you should have sent it by email or text or phone call!
Keep in mind that nothing is really private anymore. The good folks at Twitter can see your messages, not that they would, but they could. There could be a snafu in the posting of a DM, most likely a user error inside an app, that posts a supposed-to-be-back-channel to front street.