YES! It took some work, and I'm not entirely sure what made it work, but I am now writing this from my regular computer, and not my laptop!
I guess I should tell you what I did, since a couple of you really did try to help me out: First of, I switched PCI slot for the network card, but I didn't have a free slot, so I had to take out the modem (which i don't use), but that somehow made the computer not want to start because of a system failure or something like that. So I put the modem back in and used one slot that I didn't really want to use, since it was a really tight fit with the thicker graphics card right above it. Also this slot made it kind of hard do get the cable out/in since the connector for the monitor kind of blocked the possibility to push the little "pin" on the network cable... Anyway, I put the network card in that slot, and it started to blink when I put the cable in (which it did when the system failure occured too, which was why I tried another PCI slot as well). Now everything seemed to work, but I could still not access the internet. I plugged the cable from the computer to the router instead, and somehow, that did it. Possibly it could have worked without too, since it sometimes seems to take a while for the computer to realize it's on the internet, but anyway, it works now!
Sounds pretty simple, but infact the "trial and error" procedure used to discover this, took me a couple of hours (with moving the computer, connecting everything, moving cards and so on...), and I kind of missed the second half of the champions league game between Celtic and Barcelona (but I did see all the interesting stuff), but it was worth it!
Again, thanks a lot for trying to help me! Who knows, someday some other guy or girl might need this info...