You will need the Full Write Connections permission to perform this action.
There are two ways a connection can be deleted.
From the Connection
Navigate to the connection record either from the main connection listing or from the person's profile from within the Involvement > Connections tab.
Then select "Delete" from the "Actions" menu in the top right.
From the Person's Profile
You can also delete a connection right from the associated person's profile. Navigate to the profile, and click on the "Involvement" tab. Then select the "Connections" tab. Finally, select "Delete" from the drop down menu at the end of the row containing the connection you want to delete.
Regardless of which method you use, a confirmation message will appear. Click on "Yes" to finish deleting the connection.