One website won't work on one computer no matter what
Solution 1:
Things to try:
Try running
ping facebook.com
from your broken computer and from a computer that accesses it successfully. Do you get the same IP address when it looks up the hostname on both computers? Can the broken computer resolve an IP at all?Try other websites with dynamic content
Make sure the router doesn't have any type of content filtering on it that may be blocking you out...
Do you trust the users of all the other computers on your network? Someone may be doing something nasty.
Make sure your drivers are up to date. Drivers shipped on the CD that comes with the hardware are often minimally working, just enough to convince the customer that the hardware isn't dead, but they often are chock full of bugs. The bugs get fixed in subsequent driver releases which are pushed on the manufacturer's website.
The latter bullet happened to be the issue with the original poster, so I bolded it. You can see the diagnosis chat in the comments.
Googlers: welcome to driver heck ;-) Go to the manufacturer website (in the OP's case, Asus) and update your drivers!!! Some keywords:
Realtek 8111E slow Asus M5A97 ethernet slow Driver CD network slow
Solution 2:
I was having the same problem but with my twitter account. It turns out that AVG was set to block every cookie from Twitter so it won't let me follow/rt, anything. As soon as I changed that set to enable cookies it magically worked again :P