How to replace the stereo head-unit of your car:

You will need:
New head unit
New wiring harness
Dash kit
Ratchet set
Assorted Screwdrivers
Wiring Butt Connectors
Electrical tape
A wire stripper/crimper

First things first, find out how to remove your dash.  Usually the face will pull off if certain surrounding pieces are removed first.  It may require a search on-line.  

Once the dash/console facings have been removed, you should see a way to loosen the mounting of the head unit.  It is either screwed or bolted into place, in a metal box or just two slide brackets.

After the head unit is loosened pull it out far enough that you can see the wires going into the back of it.  There will be a ground wire and/or an antenna wire that are separate from the main wire clip.  Remove all of these and unclip the wire clip, this may require gentle use of pliers if not enough force can be exerted in it's attached position.

Without wire attachments the head unit should be free, pull it out and set it off to the side. Check the wire clip port for the new head unit and match the old clip to part of the new harness, and the other part of the harness to the new head unit.

If both of these match use the butt connectors and wire crimper to connect the parts of the harness.  Tape over the connections with electrical tape.

Plug both ports into their places, and replace the head-unit in the proper mounting brackets.  Sometimes fitment will be an issue, for these instances you will need to have the facing kit that corrects the fitting of the new head unit.

Fit everything back together.