Is there a way to run Windows XP from a cd (not from a hard drive)? 
I've seen some stuff on windows embedded but it's a little complicated
and it seems that it will only work on the computer that the cd was
made from.

You cannot run a standard version of XP from a CD. The files on it are
organized in a fashion that makes doing so impossible, without some
kind of additional program managing them.

The term for running an OS from a CD, at least in the Linux world, is
a LiveCD. If you are interested in making your own XP LiveCD, take a
look at the following articles:

http://www.pcquest.com/content/handson/2004/104040510.asp
http://www.pcquest.com/content/handson/2004/104050802.asp

I hope this helps.

Passive

Yes, it is possible, to a limited degree. You will need your original
XP CDs before you begin. Go to http://www.nu2.nu/pebuilder/

Bart PE offers a way to make a bootable version of 2000 or XP on CD.
This works much like a Linux LiveCD. Follow the instructions to make a
bootable CD that gives you a rudimentary Windows XP install. But first
read through the website and look at the screen shots. It is a far cry
from having a full Windows XP setup in which you can install and run
any program you want. But if you need a way to get onto any machine,
access the hard drive, and run helpful utilities, this will work.

Windows XP (Pro, Home, Embedded) can be run off a CD-R or DVD-R if you
use third party product.
Have a look at Qualystem Rescue products:
http://www.qualystem.com/en/rescue.html
It can boot Windows 9X/ME/NT/2000/XP (2003 ?) off a CD-R/DVD-R

HTH

PS: I do work for Qualystem.