I recently bought a Belkin Networking Card for my Laptop (Model
F5D5020). See description:(http://www.pricegrabber.com/search_getprod.php/masterid=438444/mode=googlecp/ut=ab4017a8ab5e3b8f)
My laptop runs Windows XP.

As I installed the card into my laptop using the latest driver from
Belkin.Com a message stoped the process with the warning "This driver
does not pass the Windows XP logo requirements, Windows XP does not
gurantee that the hardware will work on this system" ... I ignored the
warning and installed it anyway.

The network card does not work (I have tried it at more than one
internet connection) I repeated the procedure over and over -
downloading drivers from Belkin.com that say are compatible with
Windows XP, but with the same results.

The network shows up as installed in the PC. However when it shows up
on the bottom right corner of my screen (where the time is) it has a
red 'x' block. When I take the mouse over the connection, a message
saying 'Network card is unplugged' pops up. Have I not set up the card
correctly?

Am I doing something wrong, or is the network card defective?

It is a bit difficult to determine what is wrong in this situation but
there are a few things you can try

1. try another PCMCI card in the slot to see if the slot itself is working
2. try the pcmci card in another laptop 
3. check all the network settings
4. check the hardware implementation of PCMCI in your setup (see if
there are any red x's or yellow exclamation marks there)
5. by nature pcmci is plug and play , after popping the card in the
slot the pc should ask for the driver , if it doesn't there is
something wrong

If all fails go to a computer repair shop (compusa or the like) and
have them have a look at it it may cost you a few bucks but that is
better then no answer at all