On Monday,it was rolling out a feature that allows anyone to send a direct message to or receive one from anyone else. Prior to this feature, you could only DM with people who followed you (and you followed back). The new feature allows anyone that follows you the ability to send a private message, eliminating the need for you to follow them as well.
You can enable or disable the new direct message feature by logging into your account on the Web and visiting the settings page. Click here to be taken directly to it after logging in, or sign in and click on your profile photo, followed by Settings, then Security and Privacy. Scroll down to the bottom of the screen and uncheck the box (or make sure the box isn't checked) if you'd rather not receive messages from just anyone.
Additionally, when viewing a user profile on either the iPhone or Android Twitter app, you'll see a new private message icon indicating you can message the person.