"DM" on Twitter stands for "direct message." This is a feature that allows users to send short, private messages to certain other users on the website.
In the past, Twitter only allowed direct messages between two users who were both following each other on the website. Twitter later expanded the feature to allow users to DM anyone who was following them, even if the following was not mutual.
In 2013, Twitter introduced an optional setting that users can enable to allow anyone to send them direct messages. Direct messages have the same 140-character limit as regular Tweets, but unlike public Tweets, they are not accessible forever. As of 2014, Twitter shows users only the last 100 DMs exchanged.