Only admins have permission to delete contacts, and only PERSON contacts can be deleted (not companies).
To do this, go to the Contact Details page of the contact (Company OR Person views) you would like to delete. Click the "Delete" button in the top-right corner.
Company Detail View
However, if this contact (Person) is currently associated with a company (merchant, broker, funding company, etc.), you must break this association first. For example, in the example above, the contact is associated with a company called, 9 Funders. If you attempt to delete a contact that is tied to a company, you will be prompted to remove the association:

Delete a contact
To do this, click on the Company > Contacts tab for the company your contact is tied to. For example, below is the Contacts page for the company, 9 Funders. You will see a spread of all the contacts associated with this company. Simply click on the "Delete" button underneath the contact snapshot:

This only breaks the association between contact and company. In other words, the contact still exists, but no longer has a tie to this company. If you would like to continue to delete the contact entirely, return to its Contact Details page and hit the "Delete" button in the top-right corner. You will now be able to delete the contact.