Only certain users have permission to delete a contact. Permission to delete contacts is set on Organisation Preferences
Once permission to delete contacts has been enabled for your category of user, you can delete a contact from the Manage Contacts page.
Steps
Go to My Apps in the top menu
Select the Contacts App
Click the three dot button on the right next to the contact you want to delete
Select the Delete option
A confirmation popup appears
Press Yes to Delete
A popup might appear alerting you that the contact you have deleted cannot be found. This is confirmation that your contact has been deleted. When you next reload the page, you will find that the deleted contact is no longer shown.