A DxDiag, short for Direct X diagnostic, is a listing of the components of your computer. This list includes your CPU, GPU (video card), memory, recent driver updates, and other useful pieces of information about your computer. A DxDiag does not contain any sensitive information that Riot or anyone else can use against you; it is just a list of hardware and software installed on your computer.
DxDiags are used to quickly determine what system hardware or drivers may be conflicting with League of Legends. Whenever posting a technical issue on the support forums, or sending an email to firstname.lastname@example.org, it's a good idea to include this diagnostic.
Here is a short guide to obtaining a DxDiag:
Press [Windows Key] + 'R' key
Type dxdiag in the field and hit 'Ok'
The DirectX Diagnostic Tool will open. Press the "Save All Information..." button.