Definition: A direct message (DM) is a private form of communication between social media users that is only visible to the sender and recipient(s). Online businesses communicating with customers often use direct messages to clarify order details, answer complex product questions, or resolve disputes out of the public eye.
Social media plays a major role in crafting and maintaining a business' online reputation. Publicly-visible praise goes a long way to validating a brand, while criticism can have a significant negative impact on sales. Though it's important for brands to address negative criticism and not run away from it, there's no reason for long-winded communications to play out in public.
When users tweet negative feedback or post on a business Facebook page, one tried-and-true method is to respond publicly and follow up with a direct message. This makes it clear you're not hiding from the encounter and shows the complainer that you stand behind your products or services; and just as importantly, it shows this to viewers of your account.
For example, customers often use social media to report problems with their order. Reply using the medium they initiated you on — Facebook Wall, tweet, or Instagram post — and apologize for the inconvenience, stressing that an excellent customer experience is very important to the company. Then ask them to DM their order details so you can resolve the dispute out of the public eye. Other users don't want or need to see the details of every interaction; they just want to be reassured that customers are treated fairly.