(See the best response by Murrawhip.)
BYOND Build Number: Version 496.1145
Operating System: Windows 7
Video/Graphics Card: Standard VGA Graphics Adapter
Game Hub/Link: Any/All
Internet Connection Type: Wireless
Firewalls/Routers: Verizon Router Unknown type and Number

Problem Description: I have experienced this problem on past computers which have DirectX installed in and the BYOND client comes up with this message

'Hardware Rendering':
Unable to render graphics in DirectX. Falling back on software (GDI) mode. If this condition persists, you may want to update your DirectX drivers and/or dlls (BYOND requires DirectX 9 or later to take advantage of hardware rendering). Alternatively, you may disable hardware rendering through the client preferences. You can access this by hitting the F1 key to pull up the "Options & Messages" dialog and then selecting "Preferences" under the "Client" enu.

DX error message: Unable to create device

Steps to Reproduce Problem: I attempted to play League of Legends on this computer to see if it would fail there as well, and a message ame up saying "Cannot Load direct3d" which was in the package for DirectX.

Things I Did to Fix the Problem: Tried installing DirectX from this link which contains everything I need to run LoL and BYOND supposedly:
Best response
Thelavaking wrote:
Video/Graphics Card: Standard VGA Graphics Adapter

This leads me to believe that you'll need to install/update video card drivers.
The LoL support site describes the process here.
I pulled up dxdiag and it came up telling me this:

DirectDraw Acceleration: Not Available
Direct3D: Not Available
AGP Texture Acceleration: Not Available

According to everything the DirectX 11 is there with all of that stuff, and my driver IS updated. I even did a windows update. It still tells me this.
Sounds to me like your video chip doesn't support DirectX properly.
In response to Thelavaking
What video card does your computer use?

Windows 7 is not capable of keeping all video cards updated through its automatic update.
Video/Graphics Card: Standard VGA Graphics Adapter

And it used to because I have used DirectX games on this computer before.
That isn't your graphics card I'm afraid. 'Standard VGA Graphics Adapter' is the adapter name given by the built-in Windows driver, used when Windows cannot find a more suitable driver (such as an Nvidia/ATI/Intel driver). The built-in driver is a bare-bones driver, and doesn't support DirectX rendering.

My 'Display Adapter' in the Device Manager for example, is 'Intel(R) HD Graphics Family'. This is the name of the graphics driver in use, the actual graphics hardware is the Intel HD Graphics 2000 built into my Intel Core i5. If I didn't have that 'Intel(R) HD Graphics Family' driver installed, dxdiag would have said my card name was 'Standard VGA Graphics Adapter', like yours.