.dbx files are used by Microsoft's Outlook express email application and contain your email messages.
Outlook Express saves your .dbx in a folder on your computer. To find this folder, open outlook express and select the maintenance option (tools > options > maintenance). In this folder, you may find a number of .dbx files. Outlook Express creates a .dbx file for each message folder in Outlook Express. So inbox.dbx will contain emails from your inbox for example.
You'll also notice a file called folders.dbx. This file contains an index of all your message folders (and newsgroups) in Outlook express.
It is very difficult to extract individual emails from a .dbx file, however there are many applications available that are able to do this (see DBXtract link below).
Outlook Express comes bundled with most versions of the Windows operating system and is used to compose and view emails. Outlook express is probably the most common email client on the Windows platform. If you have the Microsoft Office suite, a more advanced version of Outlook Express, called Outlook is available which includes additional features such as a calendar.