So, I'm using a router as a wireless bridge, set up as the following:
[internet connection]--FiOS router--> wireless connection to wireless router--PC.
I can't seem to load any web sites, nor ping any external IPs, but when pinging I can retrieve the IP address of any website I try, so I am assuming it's not a DNS problem. In addition, all other internet services (games, torrents, IRC) continue to function perfectly.
What is my problem here? Since the computers that are directly connected to the FiOS router have no problems, I have to assume it's my bridge's problem, but I can't figure out what specifically.

Your browser is the problem, right?
Try a different browser, or look for some incorrect settings.

If that doesn't work, then check your router settings.

If that doesn't work, I think you're screwed.

I'm screwed.
Browser I've tried Chrome, Firefox, and IE, so I doubt it's that simple.
Router settings are a complicated nightmare of bad advice and early mornings. I've redone them several times as per the firmware's (dd-wrt) wiki and IRC channel's advice, but it's still a nightmare.
What's worse is the computer is in an apartment a floor above the router, and I can't just drill through the ceiling to run an Ethernet cord, so I'll have to look into getting a wireless card.

Setup an HTTP proxy on one of the other computers.

Go into command prompt, disconnect and reconnect yourself.
I forget the exact command though.

flushdns

Sounds like somewere in all your hardware, something is blocking port 80, if everything else is working.

Wait, do secure sites work?

An easy thing to try is as Mail and I said,

Type into your command prompt:

ipconfig /release

ipconfig /renew

I could also a imagine a sadisic virus that infects routers and blocks port 80.

There is a worm out there that brute forces the password on routers and loads firmware based on it's guess of what the router is.

Easiest solution:

Unplug/power down all of your devices. Computer, bridge, router, modem.
Remove all the power and data cables.
Wait 120 seconds.
Plug in data and power cables.
Power up hardware.

Chances are you just need to flush your devices.

If that doesn't work, just call your ISP. Almost no one on this forum is a qualified network technician, whereas the tech support for your ISP is full of lowly Tier 1 Help Desk Technicians.

Anything anyone else suggests in this thread is stupid (except for maybe the /release /renew suggestion)

Except DNS borking would explain it. Try connecting to an IP, like Datarealms.

if that's the case, he should just be smart like everyone else and use OpenDNS. Its up a lot more than my local ISP DNS

Damn, you guys beat me to it, I was just about to suggest he check his DNS settings.... Well, try that first, and if it doesn't work than try the unplug everything method. The unplug one means the problem is with your router. (Which is often MY own case.)