On a computer, at the top of each page, towards the right of the screen, there is your account name and a row of icons.
The two overlapping speech bubbles is the icon for DMs. Click on that and your recent messages and a button to start a new message will appear in a drop-down. If you receive a DM a green circle with a number [of unread messages] will appear pointing to the icon.
If you're using a phone, there is a messages button at the bottom of the screen. The green circle appears at the top of the screen over the three lines menu icon and the number includes all notifications, including new messages, that you have yet to view. In both versions, the number only appears until you view the list (so for messages, before you read them).