How do I remove the old cartridge from the faucet assembly??

If it's the one I am thinking, remove faucet, slide little clip out of the side, and pull out.
You will need a puller.
I used to borrow one from supply house, then bought one. Local Plumbing Supply should have tool for $10-$15.

Unless the cartridge's stuck you won't need a cartridge puller. For instructions on changing a Moen cartridge click on; Installation Help / Animated Tutorial - Installing the Moen 1225 / 1200 Cartridge (http://www.moen.com/consumersupport/installation/moen0133/moen0133.cfm)
More questions? I'm as close as a click. Tom