I do not have to mention here how important is to make a reliable backup of your data before considering doing anything described in this post !!!
If you are planning to trash or recycle your old computer, below is a good article on how to destroy your old computer hard drive... Typically removal and physical destruction of the hard drive is the fastest and easiest way to do this.
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If you are planning to sell, donate your old computer, or give it away as a gift to your relative or a friend you probably want to make sure they can boot your computer without getting new hard drive for it. Very often using restoration or operating system CD / DVD bundled with your old computer is enough to reformat your old hard drive and reinstall OS and applications as they originally came from a computer store or a manufacturer. Make sure to boot from restoration media and follow on-screen instructions selecting the most destructive restoration method where your drive is going to be low level re-formatted and brought back to factory default software. Depending on your old computer specifications whole process may take a few hours to complete so be patient.
If restoration media is no longer available or you simply prefer to give away your old computer to a family member or a friend without affecting any application or software program installed on it (still, you do want your personal data removed) your best option is to create new user account with administrative privileges, reboot computer, log on with the newly created user account, and remove all other users / accounts under Control Panel (PC) or under System Preferences (Mac). It's important to mention that you have to agree to optional removal of user settings and data when prompted by confirmation dialog. After that you will need to check individual installation directories of every 3rd party software installed on your computer (QuickBooks, Quicken , etc.) and remove all your personal data files you can find there manually. As you can already see this procedure can become lengthy and complicated enough to justify getting external help of qualified computer professional or at least somebody who knows what they are doing.
EPA approved recycling options: http://www.epa.gov/osw/conserve/materials/ecycling/donate.htm