I installed Ubuntu 12.04 on my old HP Pavilion ze1210 laptop. When I boot to Ubuntu it shows less than half of the screen; the rest is garbled and has a mixture of red and black squares. The top portion that is somewhat visible when the computer boots becomes distorted when I move the cursor (a red square) over it.
You do not have enough RAM in your HP Pavilion ze1210 laptop to install Ubuntu 12.04, and as a result you are getting visual artifact errors. Ubuntu 12.04 requires at least 384MB of RAM, but according to Google your laptop has only 256MB of RAM. Xubuntu, which is a lightweight version of Ubuntu, requires 512MB RAM, so you cannot install Xubuntu either.
So here is how to solve your problem. Your laptop meets the system requirements for Lubuntu 13.10, so install the lightweight Lubuntu 13.10. Lubuntu 13.10 has a minimalistic desktop environment that resembles Windows XP in look and feel. You can see this for yourself by visiting the Coding Studio's Lubuntu 13.10 Screenshot Tour.
UPDATE: Lubuntu 14.04 is a Long Term Support release. Security and maintenance updates will be provided for Lubuntu 14.04 until April, 2017.