You can only delete contacts if your User Type has been given permission to delete contacts.
Deleting a contact is permanent and cannot be undone.
If you are an Organisation Admin, you can choose which types of user have permission to delete contacts by granting permission to delete from the Organisation Preferences 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.