RealTek RTL8019 Network Card
 

I recently installed RedHat Linux 7.0 on a PC that previously had Windows 98 installed, however, the installation did not detect the RealTek RTL 8019 network card. From searches on the Web it appears that Linux does support this network card. I would appreciate it if anyone could help

It definitely supports the card, since I'm using it :)

Windows, linux, all OS's use drivers to access hardware, so you need a one of those. They're called modules in linux.
Now, depending on your distribution, you will get a set of precompiled ones for the most common hardware. You will have them in /lib/modules/<kernel version>/. Look for a file called something like rtl8019.o in one of the directories under there. If your dist didn't come with an RTL8019 driver then you need to get it elsewhere.

I don't quite remember if the driver comes with the linux kernel source, I think it does, so if you know how to compile the kernel or modules, you should check there. It's not that tough once you get the hang of it.
You can also check out the webpage of your NIC manufacturer and see whether they have a precompiled module, for an easier solution.
Lastly you can always check the web for that file, it's possible that someone else compiled a module like that that just might work for you.

kater@backyard:~$ lsmod | grep too
8139too 15528 1
mii 2272 0 [8139too]
kater@backyard:~$

Those are the required modules. Just 'modprobe' them and / or write them down in /etc/modules.conf