Install network card driver
http://doc.slitaz.org/en:handbook:networkconf

thanks but I've already seen that and I really couldn't figure out which module to use.

I figured out that I have a Intel® 82567 Gigabit Ethernet Controller on the computer and I downloaded the driver on the Intel Website for Linux. However I don't know how to install it. I'm not very adept yet with the bourne shell and I'm used to "make install" but of course I can't use that command.

Does anyone know what I should do?

According to google the e1000 module supports Intel® 82567 Gigabit Ethernet Controller.
Stable and Cooking both have the compiled e1000 kernel module.
Open xterm,become root
1.

root@slitaz: tazhw detect-pci

enter
This script is supposed to detect hardware and load the correct module.
It worked if it replys >Module in use : e1000
If it works go to step 3. if not step 2.
2.

root@slitaz: modprobe -v e1000

enter
redo step 1. to verify e1000 module is loaded
3.

root@slitaz:ifconfig eth0 up

enter
4.

root@slitaz: ifconfig eth0

enter
you should see Link encap:Ethernet HWaddress of your intel card listed
5.

root@slitaz: udhcpc -i eth0

enter
This requests i.p. address for eth0
If the reply is you were assigned i.p. address your connected.

To auto load e1000 module on boot open as root /etc/rcS.conf with text editor.
add e1000 to LOAD_MODULES=
and save.

Thanks! I tried this but here is my output. It looks like it won't even detect it...
I also realized the adapter is Intel(R) 82567LM-3 Gigabit Network Connection. I'm not sure if that's any different than what I posted above. Thanks so much for the help!

root@slitaz:/home/tux# tazhw detect-pci

Detected PCI devices Kernel modules
=============================================================
> Module in use:   snd_hda_intel
=============================================================

root@slitaz:/home/tux# modprobe -v e1000
Loading module e1000
root@slitaz:/home/tux# tazhw detect-pci

Detected PCI devices Kernel modules
=============================================================
> Module in use:   snd_hda_intel
=============================================================

root@slitaz:/home/tux# ifconfig eth0 up
ifconfig: SIOCGIFFLAGS: No such device

Dennis: Are you on 3.0??

@jozee

yes I'm using 3.0 although before that I was using 2.0 with the same problem.

you should need e1000e and not e1000:
http://cateee.net/lkddb/web-lkddb/E1000E.html
for reasons that i can't uderstand this driver seems to be built in in the kernel in slitaz 3.0:

root@slitaz:/home/tux# zcat /proc/config.gz | grep E1000
CONFIG_E1000=m
CONFIG_E1000E=y

so you should not need to load any module, it should work out of the box, unless this driver has some problem if it is built in the kernel.
so i would check the output of

cat /proc/net/dev | grep eth

if you can't see any eth* i would rebuild the kernel changing only CONFIG_E1000E=y to CONFIG_E1000E=m and retry modprobing e1000e

Thanks ernia!

cat /proc/net/dev | grep eth

only outputs "eth" followed by a bunch of zeros.

I hate to ask but I have never rebuilt a kernel. Is there some guide that I can look at to help me do that?

http://doc.slitaz.org/en:guides:kernel
if you have some problem i can try to help you.
the first one you could face would be an error in "make bzImage" due to initramfs. if you have an hard disk install it should be safe, until someone understand why there is such error, to disable initramfs support, it's only needed in livecd or liveusb.
may you post the output of cat /proc/net/dev | grep eth ? i'm not sure to understand what you said.

wow...that's inconvenient...Because I can't get internet on Slitaz, I cannot install slitaz-toolchain, so I have no way of compiling the kernel.

sorry about that here is the output:
root@slitaz:/home/tux# cat /proc/net/dev | grep eth
eth0: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0


so you have eth0, which means that the kernel's builtin driver at least detect you network card. now, before you recompile the kernel, you must understand if the problem is slitaz which can't get your interface up or the driver which is not working well. If you have not edited your /etc/network.conf slitaz try to get the data to configure your network card from a dhcp server. you could edit /etc/network.conf file and change DHCP from yes to no and STATIC from no to yes, then

/etc/init.d/network.sh restart

. What's the output of

ifconfig

now? Does the card have the default 192.168.0.6 address? If yes you don't have a dhcp server in your network or it refuses to give your card an address, if no you probably should try to recompile the kernel.
To do that you can download the packages.iso image, mount it as a loopback device in /media/cdrom and use /media/cdrom/packages/3.0 as your tazpkg mirror with tazpkg setup-mirror:
http://hg.slitaz.org/tazpkg/raw-file/tip/doc/tazpkg.en.html

that's weird. I don't get eth0 anymore. It's just
root@slitaz:/home/tux# cat /proc/net/dev | grep eth
eth 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Which would mean that it isn't detecting the network card. Should I try recompiling the kernel like you said then and try that out?

i would recompile the kernel, disabling builtin kernel net drivers and compiling all of them as modules, then i would try modprobing e1000e