How to delete a contact is pretty apparent... assuming you have the correct user permissions to do so, otherwise the delete button is simply not there.

Assuming you have admin permissions, you can correct this by clicking on the Admin Settings link in the upper right corner of the CRM, going under User Permissions, then clicking Manage Permissions for the affected user. Find the Contact section on the permissions window that comes up and change "Can delete" to Yes. Then hit save at the bottom and you should now find that delete button on your contact records as per the screenshot posted here earlier. Hope that helps.