How to Delete a Friend on Facebook.

Inevitably there will be someone on Facebook with whom you do not wish to communicate

• Open your Facebook profile.
• In the upper right-hand corner, click Account>>Privacy Settings.
• Go to the Block List heading. Enter your "friend's" name and click the Block button. That person will now be deleted from your profile
• To report your former friend to Facebook click Report/Block link.
• The administrators at Facebook will review your report. In extreme situations, they can ban the user.
• If your friend is sending you too many messages click Hide>>Hide Friend.

This does not help when someone has your e-mail address. 

How to keep them from getting it

You can't delete your e-mail address completely. But you can hide it from everyone. 

• Click Account>>Privacy Settings in the upper right-hand of your profile.
  
• Go to Contact Information.
  
• At the bottom of the page, you will see your e-mail address.
  
• The box next to your e-mail contains your privacy setting. Choose who will see your e-mail address in your profile. 
• Set this to Only Friends.
• Select Customize in the privacy settings.
• You will see the Custom Privacy box.
• Select the people who you want to be able to see your e-mail.
• Select people whom you do not want to see your e-mail.
• You can always set it to Only Me. If you are not comfortable giving your email address to Facebook, give them a different e-mail address.
• Create a new e-mail with any free service, like G-mail, Yahoo!, Hotmail, etc.
• In Facebook, go to Account>>Account Settings.
• Click on the E-mail heading.
• Enter your new e-mail address and click Add.

You can now remove your original e-mail address. Use your new e-mail address to sign-in. The people who used your old address will still have it. In order to cure that you'd have to drop that old address completely.