How do I check if my network card has been detected and is working?

Check if you can get out onto the net: ping google.com

If this fail's a few things could be happening.

As other's have said though, ifconfig will get you device details.

ifconfig

This should display something like:

eth0      Link encap:Ethernet  HWaddr AA:22:DD:CC:EE:FE
      inet addr:121.121.128.120  Bcast:128.146.128.255  Mask:255.255.255.0
      UP BROADCAST RUNNING MULTICAST  MTU:1500  Metric:1
      RX packets:28834210 errors:0 dropped:0 overruns:0 frame:0
      TX packets:34965403 errors:0 dropped:0 overruns:0 carrier:0
      collisions:0 txqueuelen:1000 
      RX bytes:17051041467 (15.8 GiB)  TX bytes:4435012315 (4.1 GiB)
      Memory:fc200000-fc220000 

lo        Link encap:Local Loopback  
      inet addr:127.0.0.1  Mask:255.0.0.0
      inet6 addr: ::1/128 Scope:Host
      UP LOOPBACK RUNNING  MTU:16436  Metric:1
      RX packets:6450 errors:0 dropped:0 overruns:0 frame:0
      TX packets:6450 errors:0 dropped:0 overruns:0 carrier:0
      collisions:0 txqueuelen:0 
      RX bytes:2266926 (2.1 MiB)  TX bytes:2266926 (2.1 MiB)

Notice how I have an inet addr defined. If you do not have an IP address then something is wrong.

If this is the case... most likely it would be a DHCP problem. Are you using network manager or anything to "manage" your network?

Can you ping 192.168.1.1 or ping 10.0.0.1? (Most likely your router's LAN ip is one of those)

If you can ping one of those, then you can see your router and you are not getting a IP from the router. This could be either your router DHCP not configured/setup or your system not requesting an IP from the router.

That is if you have a router.

Last thing to do:

ifdown eth0
ifup eth0