I gave up on using my Broadcom ethernet card and purchased SysConnect 9521 V2.0 network card. Unlike with the Broadcom, when I run ifconfig for eth0, I actually receive output instead of being told "no such device." I have put the IP address of my nameserver into the resolve.conf file. When I ping the router, or the IP of the DNS, it says "destination unreachable." With the Broadcom, it told me "network unreachable" so I think this is progress. It at least recognizes the network. This is the output of /sbin/route -n:

Destination: Gateway: Genmask: Flags: Metric: Ref: UseIface
192.168.1.5 0.0.0.0 255.255.255.0 U 0 0 eth0
169.254.0.0 0.0.0.0 255.255.0.0 U 0 0 eth0
127.0.0.0 0.0.0.0 255.0.0.0 U 0 0 lo
0.0.0.0 192.168.5.1 0.0.0.0 UG 0 0 eth0

My router's IP is 192.168.5.1 I have statically set the IP of eth0 to 192.168.5.3 and tried it with DHCP. Based on the way the router should assign IP addresses the IP of eth0 should be 192.168.5.3.

The output of /sbin/ifconfig eth0:
Link encap: Ethernet HWaddr: 00:00:5a:9c:A5:20
inet addr:192.168.5.3 Bcast 192.168.5.255 Mask: 255.255.255.0
inet6addr: fe80::200:5aff:fe9c:a520/64 Sco
UP BROADCAST NOTRAILERS RUNNING MULTICAST
RX packets: 0 errors: 0 dropped: 0 overruns: 0
TX packets: 0 errors: 0 dropped: 0 overruns: 0
collisions: 0 txqueuelen: 1000
RX bytes: 0 (0,0,b) TX bytes: 0 (0,0,b)
Interrupt: 17 Memory: df9fc000-0

I can't browse the internet with any browser, but at least I can access the network. Any ideas of what to try next? I'm just elated that the computer acknowledges the existence of eth0, but I'd like to do something with it too!

I'm not clear whether you're saying that you can ping the router by manually setting the IP, or not.

DHCP can fail for a number of reasons, and a machine with no IP address at all won't ping anything.

If you manually set the IP address and it still doesn't successfully ping, then check the network cable. The link light can be lit even though the cable is loose or faulty. Even professionals can forget to check the cables properly *cough*.

Web browsing requires a) IP connectivity, b) DNS (check this with the 'hosts' utility) and c) valid proxy server settings. Make sure that your router and local network are in proxy exclusions, because you can't access them through your ISP's proxy...

There is a strange thing
your ip adres is 192.168.5.3
but your router table does not create a network 192.168.5 but there is an entry 192.168.1.5 that should have been 192.168.5.0
I read that the using a dhcp client on your computer could be the source of your trouble
post the content of ifcfg-eth0 script

Boy, do I feel dumb. No, it wasn't that the Network cable wasn't fully plugged in. I had everything configured correctly and I just needed to reboot. Any discrepancies with with the information from my original post are probably just typos. So, I have just one more question: Is this more or less dumb than not having the cable connected?